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			<h1 id="firstHeading" class="firstHeading">Fourier transform</h1>
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				<p>The <b>Fourier transform</b> is a mathematical operation that decomposes a signal into its constituent frequencies. Thus the Fourier transform of a <a href="http://en.wikipedia.org/wiki/Musical_chord" title="Musical chord" class="mw-redirect">musical chord</a> is a mathematical representation of the amplitudes of the individual notes that make it up. The original signal depends on <a href="http://en.wikipedia.org/wiki/Time" title="Time">time</a>, and therefore is called the <i><a href="http://en.wikipedia.org/wiki/Time_domain" title="Time domain">time domain</a></i> representation of the signal, whereas the Fourier transform depends on frequency and is called the <i><a href="http://en.wikipedia.org/wiki/Frequency_domain" title="Frequency domain">frequency domain</a></i>
 representation of the signal. The term Fourier transform refers both to
 the frequency domain representation of the signal and the process that <a href="http://en.wikipedia.org/wiki/Transform_%28mathematics%29" title="Transform (mathematics)" class="mw-redirect">transforms</a> the signal to its frequency domain representation.</p>
<p>In mathematical terms, the Fourier transform 'transforms' one <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex</a>-valued <a href="http://en.wikipedia.org/wiki/Function_%28mathematics%29" title="Function (mathematics)">function</a> of a <a href="http://en.wikipedia.org/wiki/Real_variable" title="Real variable" class="mw-redirect">real variable</a> into another. In effect, the Fourier transform decomposes a function into <a href="http://en.wikipedia.org/wiki/Oscillation_%28mathematics%29" title="Oscillation (mathematics)">oscillatory</a> functions. The Fourier transform and its generalizations are the subject of <a href="http://en.wikipedia.org/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>. In this specific case, both the time and frequency domains are <a href="http://en.wikipedia.org/wiki/Bounded_set" title="Bounded set">unbounded</a> <a href="http://en.wikipedia.org/wiki/Linear_continuum" title="Linear continuum">linear continua</a>.
 It is possible to define the Fourier transform of a function of several
 variables, which is important for instance in the physical study of <a href="http://en.wikipedia.org/wiki/Wave_motion" title="Wave motion" class="mw-redirect">wave motion</a> and <a href="http://en.wikipedia.org/wiki/Optics" title="Optics">optics</a>. It is also possible to generalize the Fourier transform on <a href="http://en.wikipedia.org/wiki/Discrete_mathematics" title="Discrete mathematics">discrete</a> structures such as <a href="http://en.wikipedia.org/wiki/Finite_group" title="Finite group">finite groups</a>. The efficient computation of such structures, by <a href="http://en.wikipedia.org/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a>, is essential for high-speed computing.</p>
<table class="infobox">
<tbody><tr>
<th style="background: none repeat scroll 0% 0% rgb(204, 204, 255);"><strong class="selflink">Fourier transforms</strong></th>
</tr>
<tr>
<td><strong class="selflink">Continuous Fourier transform</strong></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Fourier_series" title="Fourier series">Fourier series</a></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a></td>
</tr>
<tr align="right">
<td>
<center><small><a href="http://en.wikipedia.org/wiki/List_of_Fourier-related_transforms" title="List of Fourier-related transforms">Related transforms</a></small></center>
</td>
</tr>
</tbody></table>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
 <span class="toctoggle">[<a href="#" class="internal" id="togglelink">hide</a>]</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Definition"><span class="tocnumber">1</span> <span class="toctext">Definition</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Introduction"><span class="tocnumber">2</span> <span class="toctext">Introduction</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#Properties_of_the_Fourier_transform"><span class="tocnumber">3</span> <span class="toctext">Properties of the Fourier transform</span></a>
<ul>
<li class="toclevel-2 tocsection-4"><a href="#Basic_properties"><span class="tocnumber">3.1</span> <span class="toctext">Basic properties</span></a></li>
<li class="toclevel-2 tocsection-5"><a href="#Uniform_continuity_and_the_Riemann.E2.80.93Lebesgue_lemma"><span class="tocnumber">3.2</span> <span class="toctext">Uniform continuity and the Riemann–Lebesgue lemma</span></a></li>
<li class="toclevel-2 tocsection-6"><a href="#The_Plancherel_theorem_and_Parseval.27s_theorem"><span class="tocnumber">3.3</span> <span class="toctext">The Plancherel theorem and Parseval's theorem</span></a></li>
<li class="toclevel-2 tocsection-7"><a href="#Poisson_summation_formula"><span class="tocnumber">3.4</span> <span class="toctext">Poisson summation formula</span></a></li>
<li class="toclevel-2 tocsection-8"><a href="#Convolution_theorem"><span class="tocnumber">3.5</span> <span class="toctext">Convolution theorem</span></a></li>
<li class="toclevel-2 tocsection-9"><a href="#Cross-correlation_theorem"><span class="tocnumber">3.6</span> <span class="toctext">Cross-correlation theorem</span></a></li>
<li class="toclevel-2 tocsection-10"><a href="#Eigenfunctions"><span class="tocnumber">3.7</span> <span class="toctext">Eigenfunctions</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-11"><a href="#Fourier_transform_on_Euclidean_space"><span class="tocnumber">4</span> <span class="toctext">Fourier transform on Euclidean space</span></a>
<ul>
<li class="toclevel-2 tocsection-12"><a href="#Uncertainty_principle"><span class="tocnumber">4.1</span> <span class="toctext">Uncertainty principle</span></a></li>
<li class="toclevel-2 tocsection-13"><a href="#Spherical_harmonics"><span class="tocnumber">4.2</span> <span class="toctext">Spherical harmonics</span></a></li>
<li class="toclevel-2 tocsection-14"><a href="#Restriction_problems"><span class="tocnumber">4.3</span> <span class="toctext">Restriction problems</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-15"><a href="#Generalizations"><span class="tocnumber">5</span> <span class="toctext">Generalizations</span></a>
<ul>
<li class="toclevel-2 tocsection-16"><a href="#Fourier_transform_on_other_function_spaces"><span class="tocnumber">5.1</span> <span class="toctext">Fourier transform on other function spaces</span></a></li>
<li class="toclevel-2 tocsection-17"><a href="#Fourier.E2.80.93Stieltjes_transform"><span class="tocnumber">5.2</span> <span class="toctext">Fourier–Stieltjes transform</span></a></li>
<li class="toclevel-2 tocsection-18"><a href="#Tempered_distributions"><span class="tocnumber">5.3</span> <span class="toctext">Tempered distributions</span></a></li>
<li class="toclevel-2 tocsection-19"><a href="#Locally_compact_abelian_groups"><span class="tocnumber">5.4</span> <span class="toctext">Locally compact abelian groups</span></a></li>
<li class="toclevel-2 tocsection-20"><a href="#Locally_compact_Hausdorff_space"><span class="tocnumber">5.5</span> <span class="toctext">Locally compact Hausdorff space</span></a></li>
<li class="toclevel-2 tocsection-21"><a href="#Non-abelian_groups"><span class="tocnumber">5.6</span> <span class="toctext">Non-abelian groups</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-22"><a href="#Alternatives"><span class="tocnumber">6</span> <span class="toctext">Alternatives</span></a></li>
<li class="toclevel-1 tocsection-23"><a href="#Applications"><span class="tocnumber">7</span> <span class="toctext">Applications</span></a>
<ul>
<li class="toclevel-2 tocsection-24"><a href="#Analysis_of_differential_equations"><span class="tocnumber">7.1</span> <span class="toctext">Analysis of differential equations</span></a></li>
<li class="toclevel-2 tocsection-25"><a href="#Fourier_transform_spectroscopy"><span class="tocnumber">7.2</span> <span class="toctext">Fourier transform spectroscopy</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-26"><a href="#Domain_and_range_of_the_Fourier_transform"><span class="tocnumber">8</span> <span class="toctext">Domain and range of the Fourier transform</span></a></li>
<li class="toclevel-1 tocsection-27"><a href="#Other_notations"><span class="tocnumber">9</span> <span class="toctext">Other notations</span></a></li>
<li class="toclevel-1 tocsection-28"><a href="#Other_conventions"><span class="tocnumber">10</span> <span class="toctext">Other conventions</span></a></li>
<li class="toclevel-1 tocsection-29"><a href="#Tables_of_important_Fourier_transforms"><span class="tocnumber">11</span> <span class="toctext">Tables of important Fourier transforms</span></a>
<ul>
<li class="toclevel-2 tocsection-30"><a href="#Functional_relationships"><span class="tocnumber">11.1</span> <span class="toctext">Functional relationships</span></a></li>
<li class="toclevel-2 tocsection-31"><a href="#Square-integrable_functions"><span class="tocnumber">11.2</span> <span class="toctext">Square-integrable functions</span></a></li>
<li class="toclevel-2 tocsection-32"><a href="#Distributions"><span class="tocnumber">11.3</span> <span class="toctext">Distributions</span></a></li>
<li class="toclevel-2 tocsection-33"><a href="#Two-dimensional_functions"><span class="tocnumber">11.4</span> <span class="toctext">Two-dimensional functions</span></a></li>
<li class="toclevel-2 tocsection-34"><a href="#Formulas_for_general_n-dimensional_functions"><span class="tocnumber">11.5</span> <span class="toctext">Formulas for general n-dimensional functions</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-35"><a href="#See_also"><span class="tocnumber">12</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-36"><a href="#References"><span class="tocnumber">13</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-37"><a href="#External_links"><span class="tocnumber">14</span> <span class="toctext">External links</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=1" title="Edit section: Definition">edit</a>]</span> <span class="mw-headline" id="Definition">Definition</span></h2>
<p>There are several <a href="#Other_conventions">common conventions</a> for defining the Fourier transform <img class="tex" alt="\hat{f}" src="wikipedia-Fourier_Transform_pliki/8205299e2dea57e730eb520073ec705a.png"> of an <a href="http://en.wikipedia.org/wiki/Lebesgue_integration" title="Lebesgue integration">integrable</a> function <span style="white-space:nowrap;"><i>ƒ</i> : <b>R</b> → <b>C</b></span> (<a href="#CITEREFKaiser1994">Kaiser 1994</a>). This article will use the definition:</p>
<dl>
<dd><img class="tex" alt="\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{- 2\pi i x \xi}\,dx, " src="wikipedia-Fourier_Transform_pliki/fc74da34236f93b1cf1bbecd9394597c.png"> &nbsp; for every <a href="http://en.wikipedia.org/wiki/Real_number" title="Real number">real number</a> ξ.</dd>
</dl>
<p>When the independent variable <i>x</i> represents <i>time</i> (with <a href="http://en.wikipedia.org/wiki/SI" title="SI" class="mw-redirect">SI</a> unit of <a href="http://en.wikipedia.org/wiki/Second" title="Second">seconds</a>), the transform variable <i>ξ</i>&nbsp; represents <a href="http://en.wikipedia.org/wiki/Frequency" title="Frequency">frequency</a> (in <a href="http://en.wikipedia.org/wiki/Hertz" title="Hertz">hertz</a>). Under suitable conditions, <i>ƒ</i> can be reconstructed from <img class="tex" alt="\hat f" src="wikipedia-Fourier_Transform_pliki/ed8db25ddc99da2e7e5a7e3e5c2e48cb.png"> by the <b>inverse transform</b>:</p>
<dl>
<dd><img class="tex" alt="f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi)\ e^{2 \pi i x \xi}\,d\xi, " src="wikipedia-Fourier_Transform_pliki/462af56cbf744659150d848728a03b49.png"> &nbsp; for every real number&nbsp;<i>x</i>.</dd>
</dl>
<p>For other common conventions and notations, including using the <a href="http://en.wikipedia.org/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> <a href="http://en.wikipedia.org/wiki/Omega" title="Omega"><i>ω</i></a> instead of the <a href="http://en.wikipedia.org/wiki/Frequency" title="Frequency">frequency</a> <i>ξ</i>, see <a href="http://en.wikipedia.org/wiki/Fourier_transform#Other_conventions" title="Fourier transform">Other conventions</a> and <a href="http://en.wikipedia.org/wiki/Fourier_transform#Other_notations" title="Fourier transform">Other notations</a> below. The <a href="#Fourier_transform_on_Euclidean_space">Fourier transform on Euclidean space</a> is treated separately, in which the variable <i>x</i> often represents position and <i>ξ</i> momentum.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=2" title="Edit section: Introduction">edit</a>]</span> <span class="mw-headline" id="Introduction">Introduction</span></h2>
<div class="rellink boilerplate seealso">See also: <a href="http://en.wikipedia.org/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></div>
<p>The motivation for the Fourier transform comes from the study of <a href="http://en.wikipedia.org/wiki/Fourier_series" title="Fourier series">Fourier series</a>. In the study of Fourier series, complicated functions are written as the sum of simple waves mathematically represented by <a href="http://en.wikipedia.org/wiki/Sine" title="Sine">sines</a> and <a href="http://en.wikipedia.org/wiki/Cosine" title="Cosine" class="mw-redirect">cosines</a>.
 Due to the properties of sine and cosine it is possible to recover the 
amount of each wave in the sum by an integral. In many cases it is 
desirable to use <a href="http://en.wikipedia.org/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>, which states that <i>e</i><sup>2<i>πiθ</i></sup>&nbsp;=&nbsp;cos&nbsp;2<i>πθ</i>&nbsp;+&nbsp;<i>i</i>&nbsp;sin&nbsp;2<i>πθ</i>, to write Fourier series in terms of the basic waves <i>e</i><sup>2<i>πiθ</i></sup>.
 This has the advantage of simplifying many of the formulas involved and
 providing a formulation for Fourier series that more closely resembles 
the definition followed in this article. This passage from sines and 
cosines to <a href="http://en.wikipedia.org/wiki/Complex_exponentials" title="Complex exponentials" class="mw-redirect">complex exponentials</a>
 makes it necessary for the Fourier coefficients to be complex valued. 
The usual interpretation of this complex number is that it gives both 
the <a href="http://en.wikipedia.org/wiki/Amplitude" title="Amplitude">amplitude</a> (or size) of the wave present in the function and the <a href="http://en.wikipedia.org/wiki/Phase_%28waves%29" title="Phase (waves)">phase</a> (or the initial angle) of the wave. This passage also introduces the need for negative "frequencies". If <i>θ</i> were measured in seconds then the waves <i>e</i><sup>2<i>πiθ</i></sup> and <i>e</i><sup>−2<i>πiθ</i></sup>
 would both complete one cycle per second, but they represent different 
frequencies in the Fourier transform. Hence, frequency no longer 
measures the number of cycles per unit time, but is closely related.</p>
<p>There is a close connection between the definition of Fourier series and the Fourier transform for functions <i>ƒ</i>
 which are zero outside of an interval. For such a function we can 
calculate its Fourier series on any interval that includes the interval 
where <i>ƒ</i> is not identically zero. The Fourier transform is also 
defined for such a function. As we increase the length of the interval 
on which we calculate the Fourier series, then the Fourier series 
coefficients begin to look like the Fourier transform and the sum of the
 Fourier series of <i>ƒ</i> begins to look like the inverse Fourier transform. To explain this more precisely, suppose that <i>T</i> is large enough so that the interval [−<i>T</i>/2,<i>T</i>/2] contains the interval on which <i>ƒ</i> is not identically zero. Then the <i>n</i>-th series coefficient <i>c</i><sub><i>n</i></sub> is given by<b>:</b></p>
<dl>
<dd><img class="tex" alt="c_n = \int_{-T/2}^{T/2} f(x)\ e^{-2\pi i(n/T) x} dx.\," src="wikipedia-Fourier_Transform_pliki/46aba385fd8a15c8e4839ccda1185ee7.png"></dd>
</dl>
<p>Comparing this to the definition of the Fourier transform it follows that <img class="tex" alt="c_n=\hat{f}(n/T)" src="wikipedia-Fourier_Transform_pliki/6fb744138d6a07c7235c09e3cc927bd1.png"> since <i>ƒ</i>(<i>x</i>) is zero outside [−<i>T</i>/2,<i>T</i>/2]. Thus the Fourier coefficients are just the values of the Fourier transform sampled on a grid of width 1/<i>T</i>. As <i>T</i> increases the Fourier coefficients more closely represent the Fourier transform of the function.</p>
<p>Under appropriate conditions the sum of the Fourier series of <i>ƒ</i> will equal the function <i>ƒ</i>. In other words <i>ƒ</i> can be written<b>:</b></p>
<dl>
<dd><img class="tex" alt="f(x)=\frac{1}{T}\sum_{n=-\infty}^\infty \hat{f}(n/T)\ e^{2\pi i(n/T) x} =\sum_{n=-\infty}^\infty \hat{f}(\xi_n)\ e^{2\pi i\xi_n x}\Delta\xi," src="wikipedia-Fourier_Transform_pliki/96d719730559f4399cf1ddc2ba973bbd.png"></dd>
</dl>
<p>where the last sum is simply the first sum rewritten using the definitions <i>ξ</i><sub><i>n</i></sub>&nbsp;=&nbsp;<i>n</i>/<i>T</i>, and Δ<i>ξ</i>&nbsp;=&nbsp;(<i>n</i>&nbsp;+&nbsp;1)/<i>T</i>&nbsp;−&nbsp;<i>n</i>/<i>T</i>&nbsp;=&nbsp;1/<i>T</i>.</p>
<p>This second sum is a <a href="http://en.wikipedia.org/wiki/Riemann_sum" title="Riemann sum">Riemann sum</a>, and so by letting <i>T</i>&nbsp;→&nbsp;∞
 it will converge to the integral for the inverse Fourier transform 
given in the definition section. Under suitable conditions this argument
 may be made precise (<a href="#CITEREFSteinShakarchi2003">Stein &amp; Shakarchi 2003</a>).</p>
<p>In the study of Fourier series the numbers <i>c</i><sub><i>n</i></sub> could be thought of as the "amount" of the wave in the Fourier series of <i>ƒ</i>.
 Similarly, as seen above, the Fourier transform can be thought of as a 
function that measures how much of each individual frequency is present 
in our function <i>ƒ</i>, and we can recombine these waves by using an integral (or "continuous sum") to reproduce the original function.</p>
<p><br>
The following images provide a visual illustration of how the Fourier 
transform measures whether a frequency is present in a particular 
function. The function depicted <img class="tex" alt="f(t)=\cos(6\pi t)e^{-\pi t^2}" src="wikipedia-Fourier_Transform_pliki/5ec882c4fb3bb0497730806008041571.png"> oscillates at 3 hertz (if <i>t</i>
 measures seconds) and tends quickly to 0. This function was specially 
chosen to have a real Fourier transform which can easily be plotted. The
 first image contains its graph. In order to calculate <img class="tex" alt="\hat{f}(3)" src="wikipedia-Fourier_Transform_pliki/2eadd23eef27b9b73b61dab03f4508eb.png"> we must integrate <i>e</i><sup>−2<i>πi</i>(3<i>t</i>)</sup><i>ƒ</i>(<i>t</i>).
 The second image shows the plot of the real and imaginary parts of this
 function. The real part of the integrand is almost always positive, 
this is because when <i>ƒ</i>(<i>t</i>) is negative, then the real part of <i>e</i><sup>−2<i>πi</i>(3<i>t</i>)</sup> is negative as well. Because they oscillate at the same rate, when <i>ƒ</i>(<i>t</i>) is positive, so is the real part of <i>e</i><sup>−2<i>πi</i>(3<i>t</i>)</sup>.
 The result is that when you integrate the real part of the integrand 
you get a relatively large number (in this case 0.5). On the other hand,
 when you try to measure a frequency that is not present, as in the case
 when we look at <img class="tex" alt="\hat{f}(5)" src="wikipedia-Fourier_Transform_pliki/6e7abb7621139335bb7b05bd435d3891.png">,
 the integrand oscillates enough so that the integral is very small. The
 general situation may be a bit more complicated than this, but this in 
spirit is how the Fourier transform measures how much of an individual 
frequency is present in a function <i>ƒ</i>(<i>t</i>).</p>
<ul class="gallery">
<li class="gallerybox" style="width: 155px">
<div style="width: 155px">
<div class="thumb" style="width: 150px; height: 150px;">
<div style="margin:23px auto;"><a href="http://en.wikipedia.org/wiki/File:Function_ocsillating_at_3_hertz.svg" class="image"><img alt="" src="wikipedia-Fourier_Transform_pliki/120px-Function_ocsillating_at_3_hertz.png" height="103" width="120"></a></div>
</div>
<div class="gallerytext">
<p>Original function showing oscillation 3 hertz.</p>
</div>
</div>
</li>
<li class="gallerybox" style="width: 155px">
<div style="width: 155px">
<div class="thumb" style="width: 150px; height: 150px;">
<div style="margin:23px auto;"><a href="http://en.wikipedia.org/wiki/File:Onfreq.svg" class="image"><img alt="" src="wikipedia-Fourier_Transform_pliki/120px-Onfreq.png" height="104" width="120"></a></div>
</div>
<div class="gallerytext">
<p>Real and imaginary parts of integrand for Fourier transform at 3 hertz</p>
</div>
</div>
</li>
<li class="gallerybox" style="width: 155px">
<div style="width: 155px">
<div class="thumb" style="width: 150px; height: 150px;">
<div style="margin:23px auto;"><a href="http://en.wikipedia.org/wiki/File:Offfreq.svg" class="image"><img alt="" src="wikipedia-Fourier_Transform_pliki/120px-Offfreq.png" height="103" width="120"></a></div>
</div>
<div class="gallerytext">
<p>Real and imaginary parts of integrand for Fourier transform at 5 hertz</p>
</div>
</div>
</li>
<li class="gallerybox" style="width: 155px">
<div style="width: 155px">
<div class="thumb" style="width: 150px; height: 150px;">
<div style="margin:25px auto;"><a href="http://en.wikipedia.org/wiki/File:Fourier_transform_of_oscillating_function.svg" class="image"><img alt="" src="wikipedia-Fourier_Transform_pliki/120px-Fourier_transform_of_oscillating_function.png" height="99" width="120"></a></div>
</div>
<div class="gallerytext">
<p>Fourier transform with 3 and 5 hertz labeled.</p>
</div>
</div>
</li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=3" title="Edit section: Properties of the Fourier transform">edit</a>]</span> <span class="mw-headline" id="Properties_of_the_Fourier_transform">Properties of the Fourier transform</span></h2>
<p>An <i>integrable function</i> is a function <i>ƒ</i> on the real line that is <a href="http://en.wikipedia.org/wiki/Lebesgue-measurable" title="Lebesgue-measurable" class="mw-redirect">Lebesgue-measurable</a> and satisfies</p>
<dl>
<dd><img class="tex" alt="\int_{-\infty}^\infty |f(x)| \, dx &lt; \infty." src="wikipedia-Fourier_Transform_pliki/4958351a30fae1ec802a1f0bcbe613a7.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=4" title="Edit section: Basic properties">edit</a>]</span> <span class="mw-headline" id="Basic_properties">Basic properties</span></h3>
<p>Given integrable functions <i>f</i>(<i>x</i>), <i>g</i>(<i>x</i>), and <i>h</i>(<i>x</i>) denote their Fourier transforms by <img class="tex" alt="\hat{f}(\xi)" src="wikipedia-Fourier_Transform_pliki/e919ba5bbf325d977a6309f8812c0195.png">, <img class="tex" alt="\hat{g}(\xi)" src="wikipedia-Fourier_Transform_pliki/4f50671e74e66cbdffe06075dcc6e814.png">, and <img class="tex" alt="\hat{h}(\xi)" src="wikipedia-Fourier_Transform_pliki/ed5783bb7086628a9666cb327b5613f1.png"> respectively. The Fourier transform has the following basic properties (<a href="#CITEREFPinsky2002">Pinsky 2002</a>).</p>
<dl>
<dt>Linearity</dt>
<dd>For any <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex numbers</a> <i>a</i> and <i>b</i>, if <i>h</i>(<i>x</i>)&nbsp;=&nbsp;<i>aƒ</i>(<i>x</i>)&nbsp;+&nbsp;<i>bg</i>(<i>x</i>), then  <img class="tex" alt="\hat{h}(\xi)=a\cdot \hat{f}(\xi) + b\cdot\hat{g}(\xi)." src="wikipedia-Fourier_Transform_pliki/1b0295280187db26d8732a1639212803.png"></dd>
<dt>Translation</dt>
<dd>For any <a href="http://en.wikipedia.org/wiki/Real_number" title="Real number">real number</a> <i>x</i><sub>0</sub>, if <i>h</i>(<i>x</i>)&nbsp;=&nbsp;<i>ƒ</i>(<i>x</i>&nbsp;−&nbsp;<i>x</i><sub>0</sub>), then  <img class="tex" alt="\hat{h}(\xi)= e^{-2\pi i x_0\xi }\hat{f}(\xi)." src="wikipedia-Fourier_Transform_pliki/45e97ecae150d69d5ebdb950a536ed04.png"></dd>
<dt>Modulation</dt>
<dd>For any <a href="http://en.wikipedia.org/wiki/Real_number" title="Real number">real number</a> <i>ξ</i><sub>0</sub>, if <i>h</i>(<i>x</i>)&nbsp;=&nbsp;<i>e</i><sup>2<i>πixξ</i><sub>0</sub></sup><i>ƒ</i>(<i>x</i>), then  <img class="tex" alt="\hat{h}(\xi) = \hat{f}(\xi-\xi_{0})" src="wikipedia-Fourier_Transform_pliki/239e9486236cf8208b1bd21e0ce0dace.png">.</dd>
<dt>Scaling</dt>
<dd>For a non-zero <a href="http://en.wikipedia.org/wiki/Real_number" title="Real number">real number</a> <i>a</i>, if <i>h</i>(<i>x</i>)&nbsp;=&nbsp;<i>ƒ</i>(<i>ax</i>), then  <img class="tex" alt="\hat{h}(\xi)=\frac{1}{|a|}\hat{f}\left(\frac{\xi}{a}\right)" src="wikipedia-Fourier_Transform_pliki/39b6777ad8c22e669df4c86a665efa6b.png">.&nbsp;&nbsp;&nbsp;&nbsp; The case <i>a</i>&nbsp;=&nbsp;−1 leads to the <i>time-reversal</i> property, which states: if <i>h</i>(<i>x</i>)&nbsp;=&nbsp;<i>ƒ</i>(−<i>x</i>), then  <img class="tex" alt="\hat{h}(\xi)=\hat{f}(-\xi)" src="wikipedia-Fourier_Transform_pliki/82d02b1f57f2107c1f26fae433277b62.png">.</dd>
<dt>Conjugation</dt>
<dd>If <img class="tex" alt="h(x)=\overline{f(x)}" src="wikipedia-Fourier_Transform_pliki/3358d8f2f4c6b7b1b85eeace29dca015.png">, then  <img class="tex" alt="\hat{h}(\xi) = \overline{\hat{f}(-\xi)}." src="wikipedia-Fourier_Transform_pliki/65450e041e68f42fe47e17784164ddf5.png"></dd>
<dd>In particular, if <i>ƒ</i> is real, then one has the <i>reality condition</i>  <img class="tex" alt="\hat{f}(-\xi)=\overline{\hat{f}(\xi)}." src="wikipedia-Fourier_Transform_pliki/5249ff227b5df026b3be25a948391678.png"></dd>
<dd>And if <i>ƒ</i> is purely imaginary, then  <img class="tex" alt="\hat{f}(-\xi)=-\overline{\hat{f}(\xi)}." src="wikipedia-Fourier_Transform_pliki/0acdc9f02a44dd0885639044f32f3d48.png"></dd>
<dt>Duality</dt>
<dd>If <img class="tex" alt="h(x)=\hat{f}(x)" src="wikipedia-Fourier_Transform_pliki/17bb37ab2491fd721f557c25a07ed2e9.png"> then  <img class="tex" alt="\hat{h}(\xi)= f(-\xi)." src="wikipedia-Fourier_Transform_pliki/8dc6afbc31d468b33a72466665b9431e.png"></dd>
<dt>Convolution</dt>
<dd>If <img class="tex" alt="h(x)=\left(f*g\right)(x)" src="wikipedia-Fourier_Transform_pliki/034cf01031eeb6af05a0d0ca96af142d.png">, then  <img class="tex" alt=" \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi)." src="wikipedia-Fourier_Transform_pliki/54e2488142dbc1465616168f09994733.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=5" title="Edit section: Uniform continuity and the Riemann–Lebesgue lemma">edit</a>]</span> <span class="mw-headline" id="Uniform_continuity_and_the_Riemann.E2.80.93Lebesgue_lemma">Uniform continuity and the Riemann–Lebesgue lemma</span></h3>
<div class="thumb tright">
<div class="thumbinner" style="width:222px;"><a href="http://en.wikipedia.org/wiki/File:Rectangular_function.svg" class="image"><img alt="" src="wikipedia-Fourier_Transform_pliki/220px-Rectangular_function.png" class="thumbimage" height="165" width="220"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Rectangular_function.svg" class="internal" title="Enlarge"><img src="wikipedia-Fourier_Transform_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
The <a href="http://en.wikipedia.org/wiki/Rectangular_function" title="Rectangular function">rectangular function</a> is <a href="http://en.wikipedia.org/wiki/Lebesgue_integrable" title="Lebesgue integrable" class="mw-redirect">Lebesgue integrable</a>.</div>
</div>
</div>
<div class="thumb tright">
<div class="thumbinner" style="width:222px;"><a href="http://en.wikipedia.org/wiki/File:Sinc_function_%28normalized%29.svg" class="image"><img alt="" src="wikipedia-Fourier_Transform_pliki/220px-Sinc_function_normalized.png" class="thumbimage" height="151" width="220"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Sinc_function_%28normalized%29.svg" class="internal" title="Enlarge"><img src="wikipedia-Fourier_Transform_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
The <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc function</a>, which is the Fourier transform of the rectangular function, is bounded and continuous, but not Lebesgue integrable.</div>
</div>
</div>
<p>The Fourier transform of integrable functions have additional 
properties that do not always hold. The Fourier transforms of integrable
 functions <i>ƒ</i> are <a href="http://en.wikipedia.org/wiki/Uniformly_continuous" title="Uniformly continuous" class="mw-redirect">uniformly continuous</a> and <img class="tex" alt="\|\hat{f}\|_{\infty}\leq \|f\|_1" src="wikipedia-Fourier_Transform_pliki/76daaf3042db7d5aeca2f9eb1f67258d.png"> (<a href="#CITEREFKatznelson1976">Katznelson 1976</a>). The Fourier transform of integrable functions also satisfy the <i><a href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a></i> which states that (<a href="#CITEREFSteinWeiss1971">Stein &amp; Weiss 1971</a>)</p>
<dl>
<dd><img class="tex" alt="\hat{f}(\xi)\to 0\text{ as }|\xi|\to \infty.\," src="wikipedia-Fourier_Transform_pliki/3482137c2e224e062f69879142089b23.png"></dd>
</dl>
<p>The Fourier transform <img class="tex" alt="\hat f" src="wikipedia-Fourier_Transform_pliki/ed8db25ddc99da2e7e5a7e3e5c2e48cb.png"> of an integrable function <i>ƒ</i> is bounded and continuous, but need not be integrable – for example, the Fourier transform of the <a href="http://en.wikipedia.org/wiki/Rectangular_function" title="Rectangular function">rectangular function</a>, which is a <a href="http://en.wikipedia.org/wiki/Step_function" title="Step function">step function</a> (and hence integrable) is the <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc function</a>, which is not <a href="http://en.wikipedia.org/wiki/Lebesgue_integrable" title="Lebesgue integrable" class="mw-redirect">Lebesgue integrable</a>, though it does have an improper integral: one has an analog to the <a href="http://en.wikipedia.org/wiki/Alternating_harmonic_series" title="Alternating harmonic series" class="mw-redirect">alternating harmonic series</a>, which is a convergent sum but not <a href="http://en.wikipedia.org/wiki/Absolutely_convergent" title="Absolutely convergent" class="mw-redirect">absolutely convergent</a>.</p>
<p>It is not possible in general to write the <i>inverse transform</i> as a Lebesgue integral. However, when both <i>ƒ</i> and <img class="tex" alt="\hat f" src="wikipedia-Fourier_Transform_pliki/ed8db25ddc99da2e7e5a7e3e5c2e48cb.png"> are integrable, the following inverse equality holds true for almost every <i>x</i>:</p>
<dl>
<dd><img class="tex" alt="f(x) = \int_{-\infty}^\infty \hat f(\xi) e^{2 i \pi x \xi} \, d\xi." src="wikipedia-Fourier_Transform_pliki/1c8a04991ea0c5d8cb66186597c82f98.png"></dd>
</dl>
<p>Almost everywhere, <i>ƒ</i> is equal to the continuous function given by the right-hand side. If <i>ƒ</i> is given as continuous function on the line, then equality holds for every <i>x</i>.</p>
<p>A consequence of the preceding result is that the Fourier transform is <a href="http://en.wikipedia.org/wiki/Injective" title="Injective" class="mw-redirect">injective</a> on <a href="http://en.wikipedia.org/wiki/Lp_space" title="Lp space"><i>L</i><sup>1</sup>(<b>R</b>)</a>.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=6" title="Edit section: The Plancherel theorem and Parseval's theorem">edit</a>]</span> <span class="mw-headline" id="The_Plancherel_theorem_and_Parseval.27s_theorem">The Plancherel theorem and Parseval's theorem</span></h3>
<p>Let <i>f</i>(<i>x</i>) and <i>g</i>(<i>x</i>) be integrable, and let <img class="tex" alt="\hat{f}(\xi)" src="wikipedia-Fourier_Transform_pliki/e919ba5bbf325d977a6309f8812c0195.png"> and <img class="tex" alt="\hat{g}(\xi)" src="wikipedia-Fourier_Transform_pliki/4f50671e74e66cbdffe06075dcc6e814.png"> be their Fourier transforms. If <i>f</i>(<i>x</i>) and <i>g</i>(<i>x</i>) are also <a href="http://en.wikipedia.org/wiki/Square-integrable" title="Square-integrable" class="mw-redirect">square-integrable</a>, then we have <a href="http://en.wikipedia.org/wiki/Parseval%27s_theorem" title="Parseval's theorem">Parseval's theorem</a> (<a href="#CITEREFRudin1987">Rudin 1987</a>, p. 187)<b>:</b></p>
<dl>
<dd><img class="tex" alt="\int_{-\infty}^{\infty} f(x) \overline{g(x)} \, dx = \int_{-\infty}^\infty \hat{f}(\xi) \overline{\hat{g}(\xi)} \, d\xi," src="wikipedia-Fourier_Transform_pliki/c315cc7593fef1dccbd904d531ea607d.png"></dd>
</dl>
<p>where the bar denotes <a href="http://en.wikipedia.org/wiki/Complex_conjugation" title="Complex conjugation" class="mw-redirect">complex conjugation</a>.</p>
<p>The <a href="http://en.wikipedia.org/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a>, which is equivalent to <a href="http://en.wikipedia.org/wiki/Parseval%27s_theorem" title="Parseval's theorem">Parseval's theorem</a>, states (<a href="#CITEREFRudin1987">Rudin 1987</a>, p. 186)<b>:</b></p>
<dl>
<dd><img class="tex" alt="\int_{-\infty}^\infty \left| f(x) \right|^2\, dx = \int_{-\infty}^\infty \left| \hat{f}(\xi) \right|^2\, d\xi. " src="wikipedia-Fourier_Transform_pliki/594753c0142a83bcef6ae228d16902ec.png"></dd>
</dl>
<p>The Plancherel theorem makes it possible to define the Fourier transform for functions in <i>L</i><sup>2</sup>(<b>R</b>), as described in <a href="http://en.wikipedia.org/wiki/Fourier_transform#Generalizations" title="Fourier transform">Generalizations</a>
 below. The Plancherel theorem has the interpretation in the sciences 
that the Fourier transform preserves the energy of the original 
quantity. It should be noted that depending on the author either of 
these theorems might be referred to as the Plancherel theorem or as 
Parseval's theorem.</p>
<p>See <a href="http://en.wikipedia.org/wiki/Pontryagin_duality" title="Pontryagin duality">Pontryagin duality</a> for a general formulation of this concept in the context of locally compact abelian groups.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=7" title="Edit section: Poisson summation formula">edit</a>]</span> <span class="mw-headline" id="Poisson_summation_formula">Poisson summation formula</span></h3>
<div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Poisson_summation_formula" title="Poisson summation formula">Poisson summation formula</a></div>
<p>The <a href="http://en.wikipedia.org/wiki/Poisson_summation_formula" title="Poisson summation formula">Poisson summation formula</a> provides a link between the study of Fourier transforms and Fourier Series. Given an integrable function <i>ƒ</i> we can consider the <a href="http://en.wikipedia.org/wiki/Periodic_summation" title="Periodic summation">periodic summation</a> of <i>ƒ</i> given by:</p>
<dl>
<dd><img class="tex" alt="\bar f(x)=\sum_{k\in\mathbb{Z}} f(x+k)," src="wikipedia-Fourier_Transform_pliki/384187bc8d002a1fbae683e710070231.png"></dd>
</dl>
<p>where the summation is taken over the set of all <a href="http://en.wikipedia.org/wiki/Integer" title="Integer">integers</a> <i>k</i>. The Poisson summation formula relates the Fourier series of <img class="tex" alt="\bar f" src="wikipedia-Fourier_Transform_pliki/ba7df4029f7555aa75a54841de17a30e.png"> to the Fourier transform of <i>ƒ</i>. Specifically it states that the Fourier series of <img class="tex" alt="\bar f" src="wikipedia-Fourier_Transform_pliki/ba7df4029f7555aa75a54841de17a30e.png"> is given by:</p>
<dl>
<dd><img class="tex" alt="\bar f(x) \sim \sum_{k\in\mathbb{Z}} \hat{f}(k)e^{2\pi i k x}." src="wikipedia-Fourier_Transform_pliki/70940e67f89050471aa3b8230d455f50.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=8" title="Edit section: Convolution theorem">edit</a>]</span> <span class="mw-headline" id="Convolution_theorem">Convolution theorem</span></h3>
<div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Convolution_theorem" title="Convolution theorem">Convolution theorem</a></div>
<p>The Fourier transform translates between <a href="http://en.wikipedia.org/wiki/Convolution" title="Convolution">convolution</a> and multiplication of functions. If <i>ƒ</i>(<i>x</i>) and <i>g</i>(<i>x</i>) are integrable functions with Fourier transforms <img class="tex" alt="\hat{f}(\xi)" src="wikipedia-Fourier_Transform_pliki/e919ba5bbf325d977a6309f8812c0195.png"> and <img class="tex" alt="\hat{g}(\xi)" src="wikipedia-Fourier_Transform_pliki/4f50671e74e66cbdffe06075dcc6e814.png"> respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms <img class="tex" alt="\hat{f}(\xi)" src="wikipedia-Fourier_Transform_pliki/e919ba5bbf325d977a6309f8812c0195.png"> and <img class="tex" alt="\hat{g}(\xi)" src="wikipedia-Fourier_Transform_pliki/4f50671e74e66cbdffe06075dcc6e814.png"> (under other conventions for the definition of the Fourier transform a constant factor may appear).</p>
<p>This means that if<b>:</b></p>
<dl>
<dd><img class="tex" alt="h(x) = (f*g)(x) = \int_{-\infty}^\infty f(y)g(x - y)\,dy," src="wikipedia-Fourier_Transform_pliki/53bb590e096c46a03fecf4e43d721849.png"></dd>
</dl>
<p>where ∗ denotes the convolution operation, then<b>:</b></p>
<dl>
<dd><img class="tex" alt="\hat{h}(\xi) =  \hat{f}(\xi)\cdot \hat{g}(\xi)." src="wikipedia-Fourier_Transform_pliki/54e2488142dbc1465616168f09994733.png"></dd>
</dl>
<p>In <a href="http://en.wikipedia.org/wiki/LTI_system_theory" title="LTI system theory">linear time invariant (LTI) system theory</a>, it is common to interpret <i>g</i>(<i>x</i>) as the <a href="http://en.wikipedia.org/wiki/Impulse_response" title="Impulse response">impulse response</a> of an LTI system with input <i>ƒ</i>(<i>x</i>) and output <i>h</i>(<i>x</i>), since substituting the <a href="http://en.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function">unit impulse</a> for <i>ƒ</i>(<i>x</i>) yields <i>h</i>(<i>x</i>)&nbsp;= <i>g</i>(<i>x</i>). In this case,  <img class="tex" alt="\hat{g}(\xi)" src="wikipedia-Fourier_Transform_pliki/4f50671e74e66cbdffe06075dcc6e814.png">  represents the <a href="http://en.wikipedia.org/wiki/Frequency_response" title="Frequency response">frequency response</a> of the system.</p>
<p>Conversely, if <i>ƒ</i>(<i>x</i>) can be decomposed as the product of two square integrable functions <i>p</i>(<i>x</i>) and <i>q</i>(<i>x</i>), then the Fourier transform of <i>ƒ</i>(<i>x</i>) is given by the convolution of the respective Fourier transforms <img class="tex" alt="\hat{p}(\xi)" src="wikipedia-Fourier_Transform_pliki/6af2ac5d101b1b70f267320e56b92e8e.png"> and <img class="tex" alt="\hat{q}(\xi)" src="wikipedia-Fourier_Transform_pliki/587e996d0005f24f8d4b4ad10b707e73.png">.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=9" title="Edit section: Cross-correlation theorem">edit</a>]</span> <span class="mw-headline" id="Cross-correlation_theorem">Cross-correlation theorem</span></h3>
<div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation</a></div>
<p>In an analogous manner, it can be shown that if <i>h</i>(<i>x</i>) is the <a href="http://en.wikipedia.org/wiki/Cross-correlation" title="Cross-correlation">cross-correlation</a> of <i>ƒ</i>(<i>x</i>) and <i>g</i>(<i>x</i>):</p>
<dl>
<dd><img class="tex" alt="h(x)=(f\star g)(x) = \int_{-\infty}^\infty \overline{f(y)}\,g(x+y)\,dy" src="wikipedia-Fourier_Transform_pliki/d6616c27a3eddd0d3454eeac7cca2de5.png"></dd>
</dl>
<p>then the Fourier transform of <i>h</i>(<i>x</i>) is:</p>
<dl>
<dd><img class="tex" alt="\hat{h}(\xi) = \overline{\hat{f}(\xi)}\,\hat{g}(\xi)." src="wikipedia-Fourier_Transform_pliki/f17c7a595a68acc14cca990cb6783e46.png"></dd>
</dl>
<p>As a special case, the <a href="http://en.wikipedia.org/wiki/Autocorrelation" title="Autocorrelation">autocorrelation</a> of function <i>ƒ</i>(<i>x</i>) is:</p>
<dl>
<dd><img class="tex" alt="h(x)=(f\star f)(x)=\int_{-\infty}^\infty \overline{f(y)}f(x+y)\,dy" src="wikipedia-Fourier_Transform_pliki/4cc2faf04314df14cc0f669fcc4d80be.png"></dd>
</dl>
<p>for which</p>
<dl>
<dd><img class="tex" alt="\hat{h}(\xi) = \overline{\hat{f}(\xi)}\,\hat{f}(\xi) = |\hat{f}(\xi)|^2." src="wikipedia-Fourier_Transform_pliki/15310f3e38c22e138cade316e9c7ed5b.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=10" title="Edit section: Eigenfunctions">edit</a>]</span> <span class="mw-headline" id="Eigenfunctions">Eigenfunctions</span></h3>
<p>One important choice of an orthonormal basis for <i>L</i><sup>2</sup>(<b>R</b>) is given by the Hermite functions</p>
<dl>
<dd><img class="tex" alt="{\psi}_n(x) = \frac{2^{1/4}}{\sqrt{n!}} \, e^{-\pi x^2}H_n(2x\sqrt{\pi})," src="wikipedia-Fourier_Transform_pliki/496cc5d509b8ea46ed8f0f824a749beb.png"></dd>
</dl>
<p>where <span class="texhtml"><i>H</i><sub><i>n</i></sub>(<i>x</i>)</span> are the "probabilist's" <a href="http://en.wikipedia.org/wiki/Hermite_polynomial" title="Hermite polynomial" class="mw-redirect">Hermite polynomials</a>, defined by <i>H<sub>n</sub></i>(<i>x</i>)&nbsp;= (−1)<sup><i>n</i></sup>exp(<i>x</i><sup>2</sup>/2)&nbsp;D<sup><i>n</i></sup>&nbsp;exp(−<i>x</i><sup>2</sup>/2). Under this convention for the Fourier transform, we have that</p>
<dl>
<dd><img class="tex" alt=" \hat\psi_n(\xi) = (-i)^n {\psi}_n(\xi) ." src="wikipedia-Fourier_Transform_pliki/c44a8b871696a9510053644cb78f2928.png"></dd>
</dl>
<p>In other words, the Hermite functions form a complete <a href="http://en.wikipedia.org/wiki/Orthonormal" title="Orthonormal" class="mw-redirect">orthonormal</a> system of <a href="http://en.wikipedia.org/wiki/Eigenfunctions" title="Eigenfunctions" class="mw-redirect">eigenfunctions</a> for the Fourier transform on <i>L</i><sup>2</sup>(<b>R</b>) (<a href="#CITEREFPinsky2002">Pinsky 2002</a>). However, this choice of eigenfunctions is not unique. There are only four different <a href="http://en.wikipedia.org/wiki/Eigenvalue" title="Eigenvalue" class="mw-redirect">eigenvalues</a> of the Fourier transform (±1 and ±<i>i</i>)
 and any linear combination of eigenfunctions with the same eigenvalue 
gives another eigenfunction. As a consequence of this, it is possible to
 decompose <i>L</i><sup>2</sup>(<b>R</b>) as a direct sum of four spaces <i>H</i><sub>0</sub>, <i>H</i><sub>1</sub>, <i>H</i><sub>2</sub>, and <i>H</i><sub>3</sub> where the Fourier transform acts on <i>H</i><sub><i>k</i></sub> simply by multiplication by <i>i</i><sup><i>k</i></sup>. This approach to define the Fourier transform is due to N. Wiener&nbsp;(<a href="#CITEREFDuoandikoetxea2001">Duoandikoetxea 2001</a>).
 The choice of Hermite functions is convenient because they are 
exponentially localized in both frequency and time domains, and thus 
give rise to the <a href="http://en.wikipedia.org/wiki/Fractional_Fourier_transform" title="Fractional Fourier transform">fractional Fourier transform</a> used in time-frequency analysis (<a href="#CITEREFBoashash2003">Boashash 2003</a>).</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=11" title="Edit section: Fourier transform on Euclidean space">edit</a>]</span> <span class="mw-headline" id="Fourier_transform_on_Euclidean_space">Fourier transform on Euclidean space</span></h2>
<p>The Fourier transform can be in any arbitrary number of dimensions <i>n</i>. As with the one-dimensional case there are many conventions, for an integrable function <i>ƒ</i>(<i>x</i>) this article takes the definition<b>:</b></p>
<dl>
<dd><img class="tex" alt="\hat{f}(\xi) = \mathcal{F}(f)(\xi) = \int_{\R^n} f(x) e^{-2\pi i x\cdot\xi} \, dx" src="wikipedia-Fourier_Transform_pliki/c30e545f1f9ae3bac014b9d1e2a7c483.png"></dd>
</dl>
<p>where <i>x</i> and <i>ξ</i> are <i>n</i>-dimensional <a href="http://en.wikipedia.org/wiki/Vector_%28mathematics%29" title="Vector (mathematics)" class="mw-redirect">vectors</a>, and <span style="white-space:nowrap;"><i>x</i> <b>·</b> <i>ξ</i></span> is the <a href="http://en.wikipedia.org/wiki/Dot_product" title="Dot product">dot product</a> of the vectors. The dot product is sometimes written as <img class="tex" alt="\left\langle x,\xi \right\rangle" src="wikipedia-Fourier_Transform_pliki/524397b2b69bc9533fabbde7500a9322.png">.</p>
<p>All of the basic properties listed above hold for the <i>n</i>-dimensional
 Fourier transform, as do Plancherel's and Parseval's theorem. When the 
function is integrable, the Fourier transform is still uniformly 
continuous and the <a href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a> holds. (<a href="#CITEREFSteinWeiss1971">Stein &amp; Weiss 1971</a>)</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=12" title="Edit section: Uncertainty principle">edit</a>]</span> <span class="mw-headline" id="Uncertainty_principle">Uncertainty principle</span></h3>
<div class="rellink boilerplate seealso">For more details on this topic, see <a href="http://en.wikipedia.org/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty principle</a>.</div>
<p>Generally speaking, the more concentrated <i>f</i>(<i>x</i>) is, the more spread out its Fourier transform <img class="tex" alt="\hat{f}(\xi)" src="wikipedia-Fourier_Transform_pliki/e919ba5bbf325d977a6309f8812c0195.png">  must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we "squeeze" a function in <i>x</i>, its Fourier transform "stretches out" in <i>ξ</i>. It is not possible to arbitrarily concentrate both a function and its Fourier transform.</p>
<p>The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an <b><a href="http://en.wikipedia.org/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a></b> by viewing a function and its Fourier transform as <a href="http://en.wikipedia.org/wiki/Conjugate_variables" title="Conjugate variables">conjugate variables</a> with respect to the <a href="http://en.wikipedia.org/wiki/Symplectic_form" title="Symplectic form" class="mw-redirect">symplectic form</a> on the <a href="http://en.wikipedia.org/wiki/Time%E2%80%93frequency_representation" title="Time–frequency representation">time–frequency domain</a>: from the point of view of the <a href="http://en.wikipedia.org/wiki/Linear_canonical_transformation" title="Linear canonical transformation">linear canonical transformation</a>, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the <a href="http://en.wikipedia.org/wiki/Symplectic_vector_space" title="Symplectic vector space">symplectic form</a>.</p>
<p>Suppose <i>ƒ</i>(<i>x</i>) is an integrable and <a href="http://en.wikipedia.org/wiki/Square-integrable" title="Square-integrable" class="mw-redirect">square-integrable</a> function. Without loss of generality, assume that <i>ƒ</i>(<i>x</i>) is normalized:</p>
<dl>
<dd><img class="tex" alt="\int_{-\infty}^\infty |f(x)|^2 \,dx=1." src="wikipedia-Fourier_Transform_pliki/08b886666769afa124751ad81e432cf6.png"></dd>
</dl>
<p>It follows from the <a href="http://en.wikipedia.org/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a> that <img class="tex" alt="\hat{f}(\xi)" src="wikipedia-Fourier_Transform_pliki/e919ba5bbf325d977a6309f8812c0195.png">  is also normalized.</p>
<p>The spread around <i>x</i>&nbsp;= 0 may be measured by the <i>dispersion about zero</i> (<a href="#CITEREFPinsky2002">Pinsky 2002</a>) defined by</p>
<dl>
<dd><img class="tex" alt="D_0(f)=\int_{-\infty}^\infty x^2|f(x)|^2\,dx." src="wikipedia-Fourier_Transform_pliki/c28fed2856d6b5f9ea3facdf6c205fe3.png"></dd>
</dl>
<p>In probability terms, this is the <a href="http://en.wikipedia.org/wiki/Moment_%28mathematics%29" title="Moment (mathematics)">second moment</a> of <img class="tex" alt="|f(x)|^2\,\!" src="wikipedia-Fourier_Transform_pliki/5be5bdf621f3346a292952269e37ffef.png"> about zero.</p>
<p>The Uncertainty principle states that, if <i>ƒ</i>(<i>x</i>) is absolutely continuous and the functions <i>x</i>·<i>ƒ</i>(<i>x</i>) and <i>ƒ</i>′(<i>x</i>) are square integrable, then</p>
<dl>
<dd><img class="tex" alt="D_0(f)D_0(\hat{f}) \geq \frac{1}{16\pi^2}" src="wikipedia-Fourier_Transform_pliki/cf4f17252db9a11864462880cbe1fd8e.png"> &nbsp;&nbsp;&nbsp;(<a href="#CITEREFPinsky2002">Pinsky 2002</a>).</dd>
</dl>
<p>The equality is attained only in the case <img class="tex" alt="f(x)=C_1 \, e^{{- \pi x^2}/{\sigma^2}}" src="wikipedia-Fourier_Transform_pliki/4cc5c328e7d88d4bcc970019fc67e265.png">&nbsp;&nbsp;&nbsp; (hence&nbsp;&nbsp;&nbsp;<img class="tex" alt="\quad \hat{f}(\xi)= \sigma C_1 \, e^{-\pi\sigma^2\xi^2}" src="wikipedia-Fourier_Transform_pliki/78468800a5fc562a0fc64ac5e64152fc.png">&nbsp;&nbsp;)&nbsp; where <i>σ</i>&nbsp;&gt; 0 is arbitrary and <i>C</i><sub>1</sub> is such that <i>ƒ</i> is <i>L</i><sup>2</sup>–normalized (<a href="#CITEREFPinsky2002">Pinsky 2002</a>). In other words, where <i>ƒ</i> is a (normalized) <a href="http://en.wikipedia.org/wiki/Gaussian_function" title="Gaussian function">Gaussian function</a>, centered at zero.</p>
<p>In fact, this inequality implies that:</p>
<dl>
<dd><img class="tex" alt="\left(\int_{-\infty}^\infty (x-x_0)^2|f(x)|^2\,dx\right)\left(\int_{-\infty}^\infty(\xi-\xi_0)^2|\hat{f}(\xi)|^2\,d\xi\right)\geq \frac{1}{16\pi^2}" src="wikipedia-Fourier_Transform_pliki/62acb1555c17fbb2fbee6af5d8181168.png"></dd>
</dl>
<p>for any <img class="tex" alt="x_0, \, \xi_0" src="wikipedia-Fourier_Transform_pliki/cd45d126ec9ce47eeeb68fbef6635899.png">&nbsp; in <b>R</b>&nbsp; (<a href="#CITEREFSteinShakarchi2003">Stein &amp; Shakarchi 2003</a>).</p>
<p>In <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, the <a href="http://en.wikipedia.org/wiki/Momentum" title="Momentum">momentum</a> and position <a href="http://en.wikipedia.org/wiki/Wave_function" title="Wave function">wave functions</a> are Fourier transform pairs, to within a factor of <a href="http://en.wikipedia.org/wiki/Planck%27s_constant" title="Planck's constant" class="mw-redirect">Planck's constant</a>. With this constant properly taken into account, the inequality above becomes the statement of the <a href="http://en.wikipedia.org/wiki/Heisenberg_uncertainty_principle" title="Heisenberg uncertainty principle" class="mw-redirect">Heisenberg uncertainty principle</a> (<a href="#CITEREFSteinShakarchi2003">Stein &amp; Shakarchi 2003</a>).</p>
<p>A stronger uncertainty principle is the <a href="http://en.wikipedia.org/wiki/Hirschman_uncertainty" title="Hirschman uncertainty">Hirschman uncertainty principle</a> which is expressed as:</p>
<dl>
<dd><img class="tex" alt="H(|f|^2)+H(|\hat{f}|^2)\ge \log(e/2)" src="wikipedia-Fourier_Transform_pliki/9dd0be3e87bdea881db15ad087e3bcfd.png"></dd>
</dl>
<p>where <i>H(p)</i> is the <a href="http://en.wikipedia.org/wiki/Differential_entropy" title="Differential entropy">differential entropy</a> of the <a href="http://en.wikipedia.org/wiki/Probability_density_function" title="Probability density function">probability density function</a> <i>p(x)</i>:</p>
<dl>
<dd><img class="tex" alt="H(p) = -\int_{-\infty}^\infty p(x)\log(p(x))dx" src="wikipedia-Fourier_Transform_pliki/3dc8fdb5bdc46c9fddc8b6fa0f953d42.png"></dd>
</dl>
<p>where the logarithms may be in any base which is consistent. The 
equality is attained for the normal distribution, as in the previous 
case.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=13" title="Edit section: Spherical harmonics">edit</a>]</span> <span class="mw-headline" id="Spherical_harmonics">Spherical harmonics</span></h3>
<p>Let the set of <a href="http://en.wikipedia.org/wiki/Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous</a> <a href="http://en.wikipedia.org/wiki/Harmonic_function" title="Harmonic function">harmonic</a> <a href="http://en.wikipedia.org/wiki/Polynomial" title="Polynomial">polynomials</a> of degree <i>k</i> on <b>R</b><sup><i>n</i></sup> be denoted by <b>A</b><sub><i>k</i></sub>. The set <b>A</b><sub><i>k</i></sub> consists of the <a href="http://en.wikipedia.org/wiki/Solid_spherical_harmonics" title="Solid spherical harmonics" class="mw-redirect">solid spherical harmonics</a> of degree <i>k</i>.
 The solid spherical harmonics play a similar role in higher dimensions 
to the Hermite polynomials in dimension one. Specifically, if <i>f</i>(<i>x</i>)&nbsp;=&nbsp;<i>e</i><sup>−<i>π</i>|<i>x</i>|<sup>2</sup></sup><i>P</i>(<i>x</i>) for some <i>P</i>(<i>x</i>) in <b>A</b><sub><i>k</i></sub>, then <img class="tex" alt="\hat{f}(\xi)=i^{-k}f(\xi)" src="wikipedia-Fourier_Transform_pliki/f18eef7f569e9819d8fb9614860b693d.png">. Let the set <b>H</b><sub><i>k</i></sub> be the closure in <i>L</i><sup>2</sup>(<b>R</b><sup><i>n</i></sup>) of linear combinations of functions of the form <i>f</i>(|<i>x</i>|)<i>P</i>(<i>x</i>) where <i>P</i>(<i>x</i>) is in <b>A</b><sub><i>k</i></sub>. The space <i>L</i><sup>2</sup>(<b>R</b><sup><i>n</i></sup>) is then a direct sum of the spaces <b>H</b><sub><i>k</i></sub> and the Fourier transform maps each space <b>H</b><sub><i>k</i></sub> to itself and is possible to characterize the action of the Fourier transform on each space <b>H</b><sub><i>k</i></sub> (<a href="#CITEREFSteinWeiss1971">Stein &amp; Weiss 1971</a>). Let <i>ƒ</i>(<i>x</i>)&nbsp;=&nbsp;<i>ƒ</i><sub>0</sub>(|<i>x</i>|)<i>P</i>(<i>x</i>) (with <i>P</i>(<i>x</i>) in <b>A</b><sub><i>k</i></sub>), then <img class="tex" alt="\hat{f}(\xi)=F_0(|\xi|)P(\xi)" src="wikipedia-Fourier_Transform_pliki/a5181bd552401398f0b68851f50d2ed3.png"> where</p>
<dl>
<dd><img class="tex" alt="F_0(r)=2\pi i^{-k}r^{-(n+2k-2)/2}\int_0^\infty f_0(s)J_{(n+2k-2)/2}(2\pi rs)s^{(n+2k)/2}\,ds." src="wikipedia-Fourier_Transform_pliki/4c0296f69c63039aa28e2d97920e9edf.png"></dd>
</dl>
<p>Here <i>J</i><sub>(<i>n</i>&nbsp;+&nbsp;2<i>k</i>&nbsp;−&nbsp;2)/2</sub> denotes the <a href="http://en.wikipedia.org/wiki/Bessel_function" title="Bessel function">Bessel function</a> of the first kind with order (<i>n</i>&nbsp;+&nbsp;2<i>k</i>&nbsp;−&nbsp;2)/2. When <i>k</i>&nbsp;=&nbsp;0 this gives a useful formula for the Fourier transform of a radial function (<a href="#CITEREFGrafakos2004">Grafakos 2004</a>).</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=14" title="Edit section: Restriction problems">edit</a>]</span> <span class="mw-headline" id="Restriction_problems">Restriction problems</span></h3>
<p>In higher dimensions it becomes interesting to study <i>restriction problems</i>
 for the Fourier transform. The Fourier transform of an integrable 
function is continuous and the restriction of this function to any set 
is defined. But for a square-integrable function the Fourier transform 
could be a general <i>class</i> of square integrable functions. As such, the restriction of the Fourier transform of an <i>L</i><sup>2</sup>(<b>R</b><sup><i>n</i></sup>) function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in <i>L</i><sup><i>p</i></sup> for 1&nbsp;&lt;&nbsp;<i>p</i>&nbsp;&lt;&nbsp;2. Surprisingly, it is possible in some cases to define the restriction of a Fourier transform to a set <i>S</i>, provided <i>S</i> has non-zero curvature. The case when <i>S</i> is the unit sphere in <b>R</b><sup><i>n</i></sup> is of particular interest. In this case the Tomas-<a href="http://en.wikipedia.org/wiki/Elias_Stein" title="Elias Stein" class="mw-redirect">Stein</a> restriction theorem states that the restriction of the Fourier transform to the unit sphere in <b>R</b><sup><i>n</i></sup> is a bounded operator on <i>L</i><sup><i>p</i></sup> provided 1&nbsp;≤&nbsp;<i>p</i>&nbsp;≤ <span style="white-space:nowrap;">(2<i>n</i> + 2) / (<i>n</i> + 3)</span>.</p>
<p>One notable difference between the Fourier transform in 1 dimension 
versus higher dimensions concerns the partial sum operator. Consider an 
increasing collection of measurable sets <i>E</i><sub><i>R</i></sub> indexed by <i>R</i>&nbsp;∈&nbsp;(0,∞): such as balls of radius <i>R</i> centered at the origin, or cubes of side 2<i>R</i>. For a given integrable function <i>ƒ</i>, consider the function <i>ƒ</i><sub>R</sub> defined by:</p>
<dl>
<dd><img class="tex" alt="f_R(x) = \int_{E_R}\hat{f}(\xi) e^{2\pi ix\cdot\xi}\, d\xi, \quad x \in \mathbb{R}^n." src="wikipedia-Fourier_Transform_pliki/be8fb22580aa02a4936700ab270da64c.png"></dd>
</dl>
<p>Suppose in addition that <i>ƒ</i> is in <i>L<sup>p</sup></i>(<b>R</b><sup><i>n</i></sup>). For <i>n</i>&nbsp;= 1 and <span style="white-space:nowrap;">1 &lt; <i>p</i> &lt; ∞</span>, if one takes <i>E</i><sub><i>R</i></sub>&nbsp;= (−R,&nbsp;R), then <i>ƒ</i><sub>R</sub> converges to <i>ƒ</i> in <i>L<sup>p</sup></i> as <i>R</i> tends to infinity, by the boundedness of the <a href="http://en.wikipedia.org/wiki/Hilbert_transform" title="Hilbert transform">Hilbert transform</a>. Naively one may hope the same holds true for <i>n</i>&nbsp;&gt; 1. In the case that <i>E</i><sub><i>R</i></sub> is taken to be a cube with side length <i>R</i>, then convergence still holds. Another natural candidate is the Euclidean ball <i>E</i><sub><i>R</i></sub>&nbsp;=
 {ξ&nbsp;:&nbsp;|ξ|&nbsp;&lt; R}. In order for this partial sum operator
 to converge, it is necessary that the multiplier for the unit ball be 
bounded in <i>L<sup>p</sup></i>(<b>R</b><sup><i>n</i></sup>). For <i>n</i>&nbsp;≥&nbsp;2 it is a celebrated theorem of <a href="http://en.wikipedia.org/wiki/Charles_Fefferman" title="Charles Fefferman">Charles Fefferman</a> that the multiplier for the unit ball is never bounded unless <i>p</i>&nbsp;=&nbsp;2 (<a href="#CITEREFDuoandikoetxea2001">Duoandikoetxea 2001</a>). In fact, when <span style="white-space:nowrap;"><i>p</i> ≠ 2</span>, this shows that not only may <i>ƒ</i><sub>R</sub> fail to converge to <i>ƒ</i> in <i>L<sup>p</sup></i>, but for some functions <i>ƒ</i>&nbsp;∈ <i>L<sup>p</sup></i>(<b>R</b><sup><i>n</i></sup>), <i>ƒ</i><sub>R</sub> is not even an element of <i>L<sup>p</sup></i>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=15" title="Edit section: Generalizations">edit</a>]</span> <span class="mw-headline" id="Generalizations">Generalizations</span></h2>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=16" title="Edit section: Fourier transform on other function spaces">edit</a>]</span> <span class="mw-headline" id="Fourier_transform_on_other_function_spaces">Fourier transform on other function spaces</span></h3>
<p>It is possible to extend the definition of the Fourier transform to 
other spaces of functions. Since compactly supported smooth functions 
are integrable and dense in <i>L</i><sup>2</sup>(<b>R</b>), the <a href="http://en.wikipedia.org/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a> allows us to extend the definition of the Fourier transform to general functions in <i>L</i><sup>2</sup>(<b>R</b>) by continuity arguments. Further <img class="tex" alt=" \mathcal{F}" src="wikipedia-Fourier_Transform_pliki/26afd73f8c17f310707120691ccc4a35.png">: <i>L</i><sup>2</sup>(<b>R</b>) → <i>L</i><sup>2</sup>(<b>R</b>) is a <a href="http://en.wikipedia.org/wiki/Unitary_operator" title="Unitary operator">unitary operator</a> (<a href="#CITEREFSteinWeiss1971">Stein &amp; Weiss 1971</a>, Thm. 2.3). Many of the properties remain the same for the Fourier transform. The <a href="http://en.wikipedia.org/wiki/Hausdorff%E2%80%93Young_inequality" title="Hausdorff–Young inequality">Hausdorff–Young inequality</a> can be used to extend the definition of the Fourier transform to include functions in <i>L</i><sup><i>p</i></sup>(<b>R</b>) for 1 ≤ <i>p</i> ≤ 2. Unfortunately, further extensions become more technical. The Fourier transform of functions in <i>L</i><sup><i>p</i></sup> for the range 2 &lt; <i>p</i> &lt; ∞ requires the study of distributions (<a href="#CITEREFKatznelson1976">Katznelson 1976</a>). In fact, it can be shown that there are functions in <i>L</i><sup><i>p</i></sup> with <i>p</i>&gt;2 so that the Fourier transform is not defined as a function (<a href="#CITEREFSteinWeiss1971">Stein &amp; Weiss 1971</a>).</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=17" title="Edit section: Fourier–Stieltjes transform">edit</a>]</span> <span class="mw-headline" id="Fourier.E2.80.93Stieltjes_transform">Fourier–Stieltjes transform</span></h3>
<p>The Fourier transform of a finite Borel measure <i>μ</i> on <b>R</b><sup><i>n</i></sup> is given by (<a href="#CITEREFPinsky2002">Pinsky 2002</a>):</p>
<dl>
<dd><img class="tex" alt="\hat\mu(\xi)=\int_{\mathbb{R}^n} \mathrm{e}^{-2\pi i x \cdot \xi}\,d\mu." src="wikipedia-Fourier_Transform_pliki/303b89b8ac31a6f8647a2a5128f562de.png"></dd>
</dl>
<p>This transform continues to enjoy many of the properties of the 
Fourier transform of integrable functions. One notable difference is 
that the Riemann–Lebesgue lemma fails for measures (<a href="#CITEREFKatznelson1976">Katznelson 1976</a>). In the case that <i>dμ</i>&nbsp;=&nbsp;<i>ƒ</i>(<i>x</i>)&nbsp;<i>dx</i>, then the formula above reduces to the usual definition for the Fourier transform of <i>ƒ</i>. In the case that <i>μ</i> is the probability distribution associated to a random variable <i>X</i>, the Fourier-Stieltjes transform is closely related to the <a href="http://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29" title="Characteristic function (probability theory)">characteristic function</a>, but the typical conventions in probability theory take <i>e</i><sup><i>ix·ξ</i></sup> instead of <i>e</i><sup>−2<i>πix·ξ</i></sup> (<a href="#CITEREFPinsky2002">Pinsky 2002</a>). In the case when the distribution has a <a href="http://en.wikipedia.org/wiki/Probability_density_function" title="Probability density function">probability density function</a>
 this definition reduces to the Fourier transform applied to the 
probability density function, again with a different choice of 
constants.</p>
<p>The Fourier transform may be used to give a characterization of continuous measures. <a href="http://en.wikipedia.org/wiki/Bochner%27s_theorem" title="Bochner's theorem">Bochner's theorem</a> characterizes which functions may arise as the Fourier–Stieltjes transform of a measure (<a href="#CITEREFKatznelson1976">Katznelson 1976</a>).</p>
<p>Furthermore, the <a href="http://en.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a> is not a function but it is a finite <a href="http://en.wikipedia.org/wiki/Borel_measure" title="Borel measure">Borel measure</a>. Its Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used).</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=18" title="Edit section: Tempered distributions">edit</a>]</span> <span class="mw-headline" id="Tempered_distributions">Tempered distributions</span></h3>
<div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Tempered_distributions" title="Tempered distributions" class="mw-redirect">Tempered distributions</a></div>
<p>The Fourier transform maps the space of <a href="http://en.wikipedia.org/wiki/Schwartz_space" title="Schwartz space">Schwartz functions</a> to itself, and gives a <a href="http://en.wikipedia.org/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> of the space to itself (<a href="#CITEREFSteinWeiss1971">Stein &amp; Weiss 1971</a>). Because of this it is possible to define the Fourier transform of <a href="http://en.wikipedia.org/wiki/Tempered_distributions" title="Tempered distributions" class="mw-redirect">tempered distributions</a>.
 These include all the integrable functions mentioned above, as well as 
well-behaved functions of polynomial growth and distributions of compact
 support, and have the added advantage that the Fourier transform of any
 tempered distribution is again a tempered distribution.</p>
<p>The following two facts provide some motivation for the definition of the Fourier transform of a distribution. First let <i>ƒ</i> and <i>g</i> be integrable functions, and let <img class="tex" alt="\hat{f}" src="wikipedia-Fourier_Transform_pliki/8205299e2dea57e730eb520073ec705a.png"> and <img class="tex" alt="\hat{g}" src="wikipedia-Fourier_Transform_pliki/9d82cd62f6e11a82be1ba79544daf6c0.png"> be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula (<a href="#CITEREFSteinWeiss1971">Stein &amp; Weiss 1971</a>),</p>
<dl>
<dd><img class="tex" alt="\int_{\mathbb{R}^n}\hat{f}(x)g(x)\,dx=\int_{\mathbb{R}^n}f(x)\hat{g}(x)\,dx." src="wikipedia-Fourier_Transform_pliki/4b17584a9acc1e99834efa445aa597ff.png"></dd>
</dl>
<p>Secondly, every integrable function <i>ƒ</i> defines a distribution <i>T<sub>ƒ</sub></i> by the relation</p>
<dl>
<dd><img class="tex" alt="T_f(\varphi)=\int_{\mathbb{R}^n}f(x)\varphi(x)\,dx" src="wikipedia-Fourier_Transform_pliki/ccd64006bd5ac6c4f495aa89ef5dbbc7.png">&nbsp;&nbsp;&nbsp;for all Schwartz functions <i>φ</i>.</dd>
</dl>
<p>In fact, given a distribution <i>T</i>, we define the Fourier transform by the relation</p>
<dl>
<dd><img class="tex" alt="\hat{T}(\varphi)=T(\hat{\varphi})" src="wikipedia-Fourier_Transform_pliki/f9ffabad6a819074d5bd3d45b5818e11.png">&nbsp;&nbsp;&nbsp;for all Schwartz functions <i>φ</i>.</dd>
</dl>
<p>It follows that</p>
<dl>
<dd><img class="tex" alt="\hat{T}_f=T_{\hat{f}}.\ " src="wikipedia-Fourier_Transform_pliki/f6cf024e45392d7888d160c3cb458410.png"></dd>
</dl>
<p>Distributions can be differentiated and the above mentioned 
compatibility of the Fourier transform with differentiation and 
convolution remains true for tempered distributions.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=19" title="Edit section: Locally compact abelian groups">edit</a>]</span> <span class="mw-headline" id="Locally_compact_abelian_groups">Locally compact abelian groups</span></h3>
<div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Pontryagin_duality" title="Pontryagin duality">Pontryagin duality</a></div>
<p>The Fourier transform may be generalized to any locally compact abelian group. A locally compact abelian group is an <a href="http://en.wikipedia.org/wiki/Abelian_group" title="Abelian group">abelian group</a> which is at the same time a <a href="http://en.wikipedia.org/wiki/Locally_compact" title="Locally compact" class="mw-redirect">locally compact</a> <a href="http://en.wikipedia.org/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff topological space</a>
 so that the group operations are continuous. If G is a locally compact 
abelian group, it has a translation invariant measure μ, called <a href="http://en.wikipedia.org/wiki/Haar_measure" title="Haar measure">Haar measure</a>. For a locally compact abelian group G it is possible to place a topology on the set of <a href="http://en.wikipedia.org/wiki/Character_group" title="Character group">characters</a> <img class="tex" alt="\hat G" src="wikipedia-Fourier_Transform_pliki/6dfc0faaa6690592e356d7e24800e16a.png"> so that <img class="tex" alt="\hat G" src="wikipedia-Fourier_Transform_pliki/6dfc0faaa6690592e356d7e24800e16a.png"> is also a locally compact abelian group. For a function <i>ƒ</i> in <i>L</i><sup>1</sup>(<i>G</i>) it is possible to define the Fourier transform by (<a href="#CITEREFKatznelson1976">Katznelson 1976</a>):</p>
<dl>
<dd><img class="tex" alt="\hat{f}(\xi)=\int_G \xi(x)f(x)\,d\mu\qquad\text{for any }\xi\in\hat G." src="wikipedia-Fourier_Transform_pliki/e158bec9b51263700dbe06686654d18c.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=20" title="Edit section: Locally compact Hausdorff space">edit</a>]</span> <span class="mw-headline" id="Locally_compact_Hausdorff_space">Locally compact Hausdorff space</span></h3>
<div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Gelfand_representation" title="Gelfand representation">Gelfand representation</a></div>
<p>The Fourier transform may be generalized to any locally compact 
Hausdorff space, which recovers the topology but loses the group 
structure.</p>
<p>Given a <a href="http://en.wikipedia.org/wiki/Locally_compact_space" title="Locally compact space">locally compact</a> <a href="http://en.wikipedia.org/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> <a href="http://en.wikipedia.org/wiki/Topological_space" title="Topological space">topological space</a> <i>X</i>, the space <i>A</i>=<i>C</i><sub>0</sub>(<i>X</i>) of continuous complex-valued functions on <i>X</i> which <a href="http://en.wikipedia.org/wiki/Vanish_at_infinity" title="Vanish at infinity">vanish at infinity</a> is in a natural way a commutative <a href="http://en.wikipedia.org/wiki/C*-algebra" title="C*-algebra">C*-algebra</a>, via pointwise addition, multiplication, complex conjugation, and with norm as the <a href="http://en.wikipedia.org/wiki/Uniform_norm" title="Uniform norm">uniform norm</a>. Conversely, the characters of this algebra <i>A,</i> denoted <span class="texhtml">Φ<sub><i>A</i></sub>,</span> are naturally a topological space, and can be identified with evaluation at a point of <i>x,</i> and one has an isometric isomorphism <img class="tex" alt="C_0(X) \to C_0(\Phi_A)." src="wikipedia-Fourier_Transform_pliki/b5a401a83d35582969a1cde3054fd55c.png"> In the case where <i>X</i>=<b>R</b> is the real line, this is exactly the Fourier transform.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=21" title="Edit section: Non-abelian groups">edit</a>]</span> <span class="mw-headline" id="Non-abelian_groups">Non-abelian groups</span></h3>
<p>The Fourier transform can also be defined for functions on a non-abelian group, provided that the group is <a href="http://en.wikipedia.org/wiki/Compact_space" title="Compact space">compact</a>.
 Unlike the Fourier transform on an abelian group, which is 
scalar-valued, the Fourier transform on a non-abelian group is 
operator-valued (<a href="#CITEREFHewittRoss1971">Hewitt &amp; Ross 1971</a>, Chapter 8). The Fourier transform on compact groups is a major tool in <a href="http://en.wikipedia.org/wiki/Representation_theory" title="Representation theory">representation theory</a> (<a href="#CITEREFKnapp2001">Knapp 2001</a>) and <a href="http://en.wikipedia.org/wiki/Non-commutative_harmonic_analysis" title="Non-commutative harmonic analysis" class="mw-redirect">non-commutative harmonic analysis</a>.</p>
<p>Let <i>G</i> be a compact <a href="http://en.wikipedia.org/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> <a href="http://en.wikipedia.org/wiki/Topological_group" title="Topological group">topological group</a>. Let Σ denote the collection of all isomorphism classes of finite-dimensional irreducible <a href="http://en.wikipedia.org/wiki/Unitary_representation" title="Unitary representation">unitary representations</a>, along with a definite choice of representation <i>U</i><sup>(σ)</sup> on the <a href="http://en.wikipedia.org/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> <i>H</i><sub>σ</sub> of finite dimension <i>d</i><sub>σ</sub> for each σ&nbsp;∈&nbsp;Σ. If μ is a finite <a href="http://en.wikipedia.org/wiki/Borel_measure" title="Borel measure">Borel measure</a> on <i>G</i>, then the Fourier–Stieltjes transform of μ is the operator on <i>H</i><sub>σ</sub> defined by</p>
<dl>
<dd><img class="tex" alt="\langle \hat{\mu}\xi,\eta\rangle_{H_\sigma} = \int_G \langle \overline{U}^{(\sigma)}_g\xi,\eta\rangle\,d\mu(g)" src="wikipedia-Fourier_Transform_pliki/50a70e3979b11907eab188e98668f240.png"></dd>
</dl>
<p>where <img class="tex" alt="\scriptstyle{\overline{U}^{(\sigma)}}" src="wikipedia-Fourier_Transform_pliki/20db84dadba089cccc48396871ad5f3b.png"> is the complex-conjugate representation of <i>U</i><sup>(σ)</sup> acting on <i>H</i><sub>σ</sub>. As in the abelian case, if μ is <a href="http://en.wikipedia.org/wiki/Absolutely_continuous" title="Absolutely continuous" class="mw-redirect">absolutely continuous</a> with respect to the <a href="http://en.wikipedia.org/wiki/Haar_measure" title="Haar measure">left-invariant probability measure</a> λ on <i>G</i>, then it is <a href="http://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">represented</a> as</p>
<dl>
<dd><span class="texhtml"><i>d</i>μ = <i>f</i><i>d</i>λ</span></dd>
</dl>
<p>for some <i>ƒ</i>&nbsp;∈&nbsp;<a href="http://en.wikipedia.org/wiki/Lp_space" title="Lp space">L<sup>1</sup>(λ)</a>. In this case, one identifies the Fourier transform of <i>ƒ</i> with the Fourier–Stieltjes transform of μ.</p>
<p>The mapping <img class="tex" alt="\mu\mapsto\hat{\mu}" src="wikipedia-Fourier_Transform_pliki/e498d620c684fbaa1b3f5d1ddd3c074c.png"> defines an isomorphism between the <a href="http://en.wikipedia.org/wiki/Banach_space" title="Banach space">Banach space</a> <i>M</i>(<i>G</i>) of finite Borel measures (see <a href="http://en.wikipedia.org/wiki/Rca_space" title="Rca space" class="mw-redirect">rca space</a>) and a closed subspace of the Banach space <b>C</b><sub>∞</sub>(Σ) consisting of all sequences <i>E</i>&nbsp;=&nbsp;(<i>E</i><sub>σ</sub>) indexed by Σ of (bounded) linear operators <i>E</i><sub>σ</sub>&nbsp;:&nbsp;<i>H</i><sub>σ</sub>&nbsp;→&nbsp;<i>H</i><sub>σ</sub> for which the norm</p>
<dl>
<dd><img class="tex" alt="\|E\| = \sup_{\sigma\in\Sigma}\|E_\sigma\|" src="wikipedia-Fourier_Transform_pliki/eb8e056dfde7da87e9f0872859633332.png"></dd>
</dl>
<p>is finite. The "<a href="http://en.wikipedia.org/wiki/Convolution_theorem" title="Convolution theorem">convolution theorem</a>" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isomorphism of <a href="http://en.wikipedia.org/wiki/C*_algebra" title="C* algebra" class="mw-redirect">C<sup>*</sup> algebras</a> into a subspace of <b>C</b><sub>∞</sub>(Σ), in which <i>M</i>(<i>G</i>) is equipped with the product given by <a href="http://en.wikipedia.org/wiki/Convolution" title="Convolution">convolution</a> of measures and <b>C</b><sub>∞</sub>(Σ) the product given by multiplication of operators in each index σ.</p>
<p>The <a href="http://en.wikipedia.org/wiki/Peter-Weyl_theorem" title="Peter-Weyl theorem" class="mw-redirect">Peter-Weyl theorem</a> holds, and a version of the Fourier inversion formula (<a href="http://en.wikipedia.org/wiki/Plancherel%27s_theorem" title="Plancherel's theorem" class="mw-redirect">Plancherel's theorem</a>) follows: if <i>ƒ</i>&nbsp;∈&nbsp;L<sup>2</sup>(<i>G</i>), then</p>
<dl>
<dd><img class="tex" alt="f(g) = \sum_{\sigma\in\Sigma} d_\sigma \operatorname{tr}(\hat{f}(\sigma)U^{(\sigma)}_g)" src="wikipedia-Fourier_Transform_pliki/d4bf735208bc7cb6e5ab8f60c5c845a8.png"></dd>
</dl>
<p>where the summation is understood as convergent in the L<sup>2</sup> sense.</p>
<p>The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of <a href="http://en.wikipedia.org/wiki/Noncommutative_geometry" title="Noncommutative geometry">noncommutative geometry</a>.<sup class="Template-Fact" title="This claim needs references to reliable sources from May 2009" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a></i>]</sup> In this context, a categorical generalization of the Fourier transform to noncommutative groups is <a href="http://en.wikipedia.org/wiki/Tannaka-Krein_duality" title="Tannaka-Krein duality" class="mw-redirect">Tannaka-Krein duality</a>,
 which replaces the group of characters with the category of 
representations. However, this loses the connection with harmonic 
functions.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=22" title="Edit section: Alternatives">edit</a>]</span> <span class="mw-headline" id="Alternatives">Alternatives</span></h2>
<p>In <a href="http://en.wikipedia.org/wiki/Signal_processing" title="Signal processing">signal processing</a> terms, a function (of time) is a representation of a signal with perfect <i>time resolution,</i> but no frequency information, while the Fourier transform has perfect <i>frequency resolution,</i>
 but no time information: the magnitude of the Fourier transform at a 
point is how much frequency content there is, but location is only given
 by phase (argument of the Fourier transform at a point), and standing 
waves are not localized in time – a sine wave continues out to infinity,
 without decaying. This limits the usefulness of the Fourier transform 
for analyzing signals that are localized in time, notably <a href="http://en.wikipedia.org/w/index.php?title=Transient_%28acoustics%29&amp;action=edit&amp;redlink=1" class="new" title="Transient (acoustics) (page does not exist)">transients</a>, or any signal of finite extent.</p>
<p>As alternatives to the Fourier transform, in <a href="http://en.wikipedia.org/wiki/Time-frequency_analysis" title="Time-frequency analysis" class="mw-redirect">time-frequency analysis</a>,
 one uses time-frequency transforms or time-frequency distributions to 
represent signals in a form that has some time information and some 
frequency information – by the uncertainty principle, there is a 
trade-off between these. These can be generalizations of the Fourier 
transform, such as the <a href="http://en.wikipedia.org/wiki/Short-time_Fourier_transform" title="Short-time Fourier transform">short-time Fourier transform</a> or <a href="http://en.wikipedia.org/wiki/Fractional_Fourier_transform" title="Fractional Fourier transform">fractional Fourier transform</a>, or can use different functions to represent signals, as in <a href="http://en.wikipedia.org/wiki/Wavelet_transforms" title="Wavelet transforms" class="mw-redirect">wavelet transforms</a> and <a href="http://en.wikipedia.org/wiki/Chirplet_transforms" title="Chirplet transforms" class="mw-redirect">chirplet transforms</a>, with the wavelet analog of the (continuous) Fourier transform being the <a href="http://en.wikipedia.org/wiki/Continuous_wavelet_transform" title="Continuous wavelet transform">continuous wavelet transform</a>. (<a href="#CITEREFBoashash2003">Boashash 2003</a>). For a variable time and frequency resolution, the <a href="http://en.wikipedia.org/w/index.php?title=De_Groot_Fourier_Transform&amp;action=edit&amp;redlink=1" class="new" title="De Groot Fourier Transform (page does not exist)">De Groot Fourier Transform</a> can be considered.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=23" title="Edit section: Applications">edit</a>]</span> <span class="mw-headline" id="Applications">Applications</span></h2>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=24" title="Edit section: Analysis of differential equations">edit</a>]</span> <span class="mw-headline" id="Analysis_of_differential_equations">Analysis of differential equations</span></h3>
<p>Fourier transforms and the closely related <a href="http://en.wikipedia.org/wiki/Laplace_transform" title="Laplace transform">Laplace transforms</a> are widely used in solving <a href="http://en.wikipedia.org/wiki/Differential_equations" title="Differential equations" class="mw-redirect">differential equations</a>. The Fourier transform is compatible with <a href="http://en.wikipedia.org/wiki/Derivative" title="Derivative">differentiation</a> in the following sense: if <i>f</i>(<i>x</i>) is a differentiable function with Fourier transform <img class="tex" alt="\hat{f}(\xi)" src="wikipedia-Fourier_Transform_pliki/e919ba5bbf325d977a6309f8812c0195.png">, then the Fourier transform of its derivative is given by <img class="tex" alt="2\pi i\xi\hat{f}(\xi)" src="wikipedia-Fourier_Transform_pliki/da1df0c255ea99720b2d8bdf1ce9a1f6.png">.
 This can be used to transform differential equations into algebraic 
equations. Note that this technique only applies to problems whose 
domain is the whole set of real numbers. By extending the Fourier 
transform to functions of several variables <a href="http://en.wikipedia.org/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> with domain <b>R</b><sup>n</sup> can also be translated into algebraic equations.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=25" title="Edit section: Fourier transform spectroscopy">edit</a>]</span> <span class="mw-headline" id="Fourier_transform_spectroscopy">Fourier transform spectroscopy</span></h3>
<div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Fourier_transform_spectroscopy" title="Fourier transform spectroscopy">Fourier transform spectroscopy</a></div>
<p>The Fourier transform is also used in <a href="http://en.wikipedia.org/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">nuclear magnetic resonance</a> (NMR) and in other kinds of <a href="http://en.wikipedia.org/wiki/Spectroscopy" title="Spectroscopy">spectroscopy</a>, e.g. infrared (<a href="http://en.wikipedia.org/wiki/Fourier_transform_infrared_spectroscopy" title="Fourier transform infrared spectroscopy">FTIR</a>).
 In NMR an exponentially-shaped free induction decay (FID) signal is 
acquired in the time domain and Fourier-transformed to a Lorentzian 
line-shape in the frequency domain. The Fourier transform is also used 
in <a href="http://en.wikipedia.org/wiki/Magnetic_resonance_imaging" title="Magnetic resonance imaging">magnetic resonance imaging</a> (MRI) and <a href="http://en.wikipedia.org/wiki/Mass_spectrometry" title="Mass spectrometry">mass spectrometry</a>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=26" title="Edit section: Domain and range of the Fourier transform">edit</a>]</span> <span class="mw-headline" id="Domain_and_range_of_the_Fourier_transform">Domain and range of the Fourier transform</span></h2>
<p>It is often desirable to have the most general domain for the Fourier
 transform as possible. The definition of Fourier transform as an 
integral naturally restricts the domain to the space of integrable 
functions. Unfortunately, there is no simple characterizations of which 
functions are Fourier transforms of integrable functions (<a href="#CITEREFSteinWeiss1971">Stein &amp; Weiss 1971</a>).
 It is possible to extend the domain of the Fourier transform in various
 ways, as discussed in generalizations above. The following list details
 some of the more common domains and ranges on which the Fourier 
transform is defined.</p>
<ul>
<li>The space of <a href="http://en.wikipedia.org/wiki/Schwartz_function" title="Schwartz function" class="mw-redirect">Schwartz functions</a>
 is closed under the Fourier transform. Schwartz functions are rapidly 
decaying functions and do not include all functions which are relevant 
for the Fourier transform. More details may be found in (<a href="#CITEREFSteinWeiss1971">Stein &amp; Weiss 1971</a>).</li>
</ul>
<ul>
<li>The space <i>L</i><sup><i>p</i></sup> maps into the space <i>L</i><sup><i>q</i></sup>, where 1/<i>p</i>&nbsp;+ 1/<i>q</i>&nbsp;= 1 and 1&nbsp;≤&nbsp;<i>p</i>&nbsp;≤&nbsp;2 (<a href="http://en.wikipedia.org/wiki/Hausdorff%E2%80%93Young_inequality" title="Hausdorff–Young inequality">Hausdorff–Young inequality</a>).</li>
</ul>
<ul>
<li>In particular, the space <i>L</i><sup>2</sup> is closed under the Fourier transform, but here the Fourier transform is no longer defined by integration.</li>
</ul>
<ul>
<li>The space <i>L</i><sup>1</sup> of Lebesgue integrable functions maps into <i>C</i><sub>0</sub>, the space of continuous functions that tend to zero at infinity – not just into the space <img class="tex" alt="L^\infty" src="wikipedia-Fourier_Transform_pliki/a777d3201f2b456889286a62549b8d20.png"> of bounded functions (the <a href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a>).</li>
</ul>
<ul>
<li>The set of <a href="http://en.wikipedia.org/wiki/Tempered_distributions" title="Tempered distributions" class="mw-redirect">tempered distributions</a>
 is closed under the Fourier transform. Tempered distributions are also a
 form of generalization of functions. It is in this generality that one 
can define the Fourier transform of objects like the <a href="http://en.wikipedia.org/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a>.</li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=27" title="Edit section: Other notations">edit</a>]</span> <span class="mw-headline" id="Other_notations">Other notations</span></h2>
<p>Other common notations for <img class="tex" alt=" \hat{f}(\xi)" src="wikipedia-Fourier_Transform_pliki/e919ba5bbf325d977a6309f8812c0195.png"> are these:</p>
<p><img class="tex" alt="\tilde{f}(\xi),\  \tilde{f}(\omega),\  F(\xi),\  \mathcal{F}\left(f\right)(\xi),\  \left(\mathcal{F}f\right)(\xi),\  \mathcal{F}(f),\  \mathcal F(\omega),\  \mathcal F(j\omega),\  \mathcal{F}\{f\},\  \mathcal{F} \left(f(t)\right)" src="wikipedia-Fourier_Transform_pliki/b9a9858f415e2f609492004433d82863.png"></p>
<p>Though less commonly other notations are used. Denoting the Fourier 
transform by a capital letter corresponding to the letter of function 
being transformed (such as <i>f</i>(<i>x</i>) and <i>F</i>(<i>ξ</i>)) is especially common in the sciences and engineering. In electronics, the omega (<i>ω</i>) is often used instead of <i>ξ</i> due to its interpretation as angular frequency, sometimes it is written as <i>F</i>(<i>jω</i>), where <i>j</i> is the <a href="http://en.wikipedia.org/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, to indicate its relationship with the <a href="http://en.wikipedia.org/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a>, and sometimes it is written informally as <i>F</i>(2<i>πf</i>) in order to use ordinary frequency.</p>
<p>The interpretation of the complex function <img class="tex" alt="\hat{f}(\xi)" src="wikipedia-Fourier_Transform_pliki/e919ba5bbf325d977a6309f8812c0195.png"> may be aided by expressing it in <a href="http://en.wikipedia.org/wiki/Polar_coordinate" title="Polar coordinate" class="mw-redirect">polar coordinate</a> form<b>:</b> &nbsp; <img class="tex" alt="\hat{f}(\xi)=A(\xi)e^{i\varphi(\xi)}" src="wikipedia-Fourier_Transform_pliki/c179401c784dd79fbf982fefbb52fea3.png"> in terms of the two real functions <i>A</i>(<i>ξ</i>) and φ(<i>ξ</i>) where<b>:</b></p>
<dl>
<dd><img class="tex" alt="A(\xi) = |\hat{f}(\xi)|, \, " src="wikipedia-Fourier_Transform_pliki/4c476d9022091b9592e78c73560b165a.png"></dd>
</dl>
<p>is the <a href="http://en.wikipedia.org/wiki/Amplitude" title="Amplitude">amplitude</a> and</p>
<dl>
<dd><img class="tex" alt="\varphi (\xi) = \arg \big( \hat{f}(\xi) \big), " src="wikipedia-Fourier_Transform_pliki/da17ffcab100ec580485a942f8200077.png"> &nbsp;</dd>
</dl>
<p>is the <a href="http://en.wikipedia.org/wiki/Phase_%28waves%29" title="Phase (waves)">phase</a> (see <a href="http://en.wikipedia.org/wiki/Arg_%28mathematics%29" title="Arg (mathematics)" class="mw-redirect">arg function</a>).</p>
<p>Then the inverse transform can be written<b>:</b></p>
<dl>
<dd><img class="tex" alt="f(x) = \int  _{-\infty}^{\infty} A(\xi)\ e^{ i(2\pi \xi x +\varphi (\xi))}\,d\xi," src="wikipedia-Fourier_Transform_pliki/9705360e623eea185a832836f5956bcf.png"></dd>
</dl>
<p>which is a recombination of all the <b>frequency components</b> of <i>ƒ</i>(<i>x</i>). Each component is a complex <a href="http://en.wikipedia.org/wiki/Sinusoid" title="Sinusoid" class="mw-redirect">sinusoid</a> of the form <i>e</i><sup>2<i>πixξ</i></sup>&nbsp; whose <a href="http://en.wikipedia.org/wiki/Amplitude" title="Amplitude">amplitude</a> is <i>A</i>(<i>ξ</i>) and whose initial <a href="http://en.wikipedia.org/wiki/Phase_angle" title="Phase angle">phase angle</a> (at <i>x</i>&nbsp;=&nbsp;0) is <i>φ</i>(<i>ξ</i>).</p>
<p>The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denoted <img class="tex" alt="\mathcal{F}" src="wikipedia-Fourier_Transform_pliki/26afd73f8c17f310707120691ccc4a35.png"> and <img class="tex" alt="\mathcal{F}(f)" src="wikipedia-Fourier_Transform_pliki/3e72ce89c3bf417c425164f9f0ccb58f.png"> is used to denote the Fourier transform of the function <i>f</i>. This mapping is linear, which means that <img class="tex" alt="\mathcal{F}" src="wikipedia-Fourier_Transform_pliki/26afd73f8c17f310707120691ccc4a35.png">
 can also be seen as a linear transformation on the function space and 
implies that the standard notation in linear algebra of applying a 
linear transformation to a vector (here the function <i>f</i>) can be used to write <img class="tex" alt="\mathcal{F} f" src="wikipedia-Fourier_Transform_pliki/834102942928839a0467c71d278c8dd5.png"> instead of <img class="tex" alt="\mathcal{F}(f)" src="wikipedia-Fourier_Transform_pliki/3e72ce89c3bf417c425164f9f0ccb58f.png">.
 Since the result of applying the Fourier transform is again a function,
 we can be interested in the value of this function evaluated at the 
value <i>ξ</i> for its variable, and this is denoted either as <img class="tex" alt="\mathcal{F}(f)(\xi)" src="wikipedia-Fourier_Transform_pliki/2fa016d36dbabe3be9aafbd2c5e9738a.png"> or as <img class="tex" alt="(\mathcal{F} f)(\xi)" src="wikipedia-Fourier_Transform_pliki/157018df1eae241c4ca96c842aa4e0b8.png">. Notice that in the former case, it is implicitly understood that <img class="tex" alt="\mathcal{F}" src="wikipedia-Fourier_Transform_pliki/26afd73f8c17f310707120691ccc4a35.png"> is applied first to <i>f</i> and then the resulting function is evaluated at <i>ξ</i>, not the other way around.</p>
<p>In mathematics and various applied sciences it is often necessary to distinguish between a function <i>f</i> and the value of <i>f</i> when its variable equals <i>x</i>, denoted <i>f</i>(<i>x</i>). This means that a notation like <img class="tex" alt="\mathcal{F}(f(x))" src="wikipedia-Fourier_Transform_pliki/f5dbd6e66239b0401bbefb1e3336b143.png"> formally can be interpreted as the Fourier transform of the values of <i>f</i> at <i>x</i>.
 Despite this flaw, the previous notation appears frequently, often when
 a particular function or a function of a particular variable is to be 
transformed. For example, <img class="tex" alt="\mathcal{F}( \mathrm{rect}(x) ) = \mathrm{sinc}(\xi)" src="wikipedia-Fourier_Transform_pliki/1bdb6026956531cbe90dd740defb3130.png"> is sometimes used to express that the Fourier transform of a rectangular function is a sinc function, or <img class="tex" alt="\mathcal{F}(f(x+x_{0})) = \mathcal{F}(f(x)) e^{2\pi i \xi x_{0}}" src="wikipedia-Fourier_Transform_pliki/1b3e933fa536f3a97f4b2994bfd259db.png">
 is used to express the shift property of the Fourier transform. Notice,
 that the last example is only correct under the assumption that the 
transformed function is a function of <i>x</i>, not of <i>x</i><sub>0</sub>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=28" title="Edit section: Other conventions">edit</a>]</span> <span class="mw-headline" id="Other_conventions">Other conventions</span></h2>
<p>The Fourier transform can also be written in terms of <a href="http://en.wikipedia.org/wiki/Angular_frequency" title="Angular frequency">angular frequency</a><b>:</b> &nbsp; <i>ω</i> = <i>2πξ</i> whose units are <a href="http://en.wikipedia.org/wiki/Radians" title="Radians" class="mw-redirect">radians</a> per second.</p>
<p>The substitution <i>ξ</i> = <i>ω</i>/(2π) into the formulas above produces this convention<b>:</b></p>
<dl>
<dd><img class="tex" alt="\hat{f}(\omega) = \int_{\mathbb{R}^n} f(x) e^{- i\omega\cdot x}\,dx." src="wikipedia-Fourier_Transform_pliki/3ff22f29daed6474ed2aba90a5a91a43.png"></dd>
</dl>
<p>Under this convention, the inverse transform becomes:</p>
<dl>
<dd><img class="tex" alt="f(x) = \frac{1}{(2\pi)^n} \int_{\mathbb{R}^n} \hat{f}(\omega)e^{ i\omega \cdot x}\,d\omega. " src="wikipedia-Fourier_Transform_pliki/86ba99a5ecc0b74461122cf28ed9cc5a.png"></dd>
</dl>
<p>Unlike the convention followed in this article, when the Fourier transform is defined this way, it is no longer a <a href="http://en.wikipedia.org/wiki/Unitary_transformation" title="Unitary transformation">unitary transformation</a> on <i>L</i><sup>2</sup>(<b>R</b><sup><i>n</i></sup>). There is also less symmetry between the formulas for the Fourier transform and its inverse.</p>
<p>Another convention is to split the factor of (2<i>π</i>)<sup><i>n</i></sup> evenly between the Fourier transform and its inverse, which leads to definitions<b>:</b></p>
<dl>
<dd><img class="tex" alt=" \hat{f}(\omega) = \frac{1}{(2\pi)^{n/2}} \int_{\mathbb{R}^n} f(x) e^{- i\omega\cdot x}\,dx " src="wikipedia-Fourier_Transform_pliki/ad13c360125dac7f88d90e6c57d45758.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="f(x) = \frac{1}{(2\pi)^{n/2}} \int_{\mathbb{R}^n} \hat{f}(\omega) e^{ i\omega \cdot x}\,d\omega. " src="wikipedia-Fourier_Transform_pliki/56f39de07f0160bdda3ce117efd318e2.png"></dd>
</dl>
<p>Under this convention, the Fourier transform is again a unitary transformation on <i>L</i><sup>2</sup>(<b>R</b><sup><i>n</i></sup>). It also restores the symmetry between the Fourier transform and its inverse.</p>
<p>Variations of all three conventions can be created by conjugating the complex-exponential <a href="http://en.wikipedia.org/wiki/Kernel_%28mathematics%29#In_integral_calculus" title="Kernel (mathematics)">kernel</a>
 of both the forward and the reverse transform. The signs must be 
opposites. Other than that, the choice is (again) a matter of 
convention.</p>
<table class="wikitable">
<caption>Summary of popular forms of the Fourier transform</caption>
<tbody><tr>
<th>ordinary frequency <i>ξ</i> (hertz)</th>
<th>unitary</th>
<td><img class="tex" alt="\displaystyle \hat{f}_1(\xi)\ \stackrel{\mathrm{def}}{=}\ \int_{\mathbb{R}^n} f(x) e^{-2 \pi i x\cdot\xi}\, dx = \hat{f}_2(2 \pi \xi)=(2 \pi)^{n/2}\hat{f}_3(2 \pi \xi) " src="wikipedia-Fourier_Transform_pliki/e0c0bc479d5d0683b735748f1d3a0227.png"><br>
<p><img class="tex" alt="\displaystyle f(x) = \int_{\mathbb{R}^n} \hat{f}_1(\xi) e^{2 \pi i  x\cdot \xi}\, d\xi \ " src="wikipedia-Fourier_Transform_pliki/aca5ebf1cb9864202833b2a738a16504.png"></p>
</td>
</tr>
<tr>
<th rowspan="2">angular frequency <i>ω</i> (rad/s)</th>
<th>non-unitary</th>
<td><img class="tex" alt="\displaystyle \hat{f}_2(\omega) \ \stackrel{\mathrm{def}}{=}\int_{\mathbb{R}^n} f(x) e^{-i\omega\cdot x} \, dx \ = \hat{f}_1 \left ( \frac{\omega}{2 \pi} \right ) = (2 \pi)^{n/2}\ \hat{f}_3(\omega) " src="wikipedia-Fourier_Transform_pliki/c3ee92c7a61738780530ed0925bb823f.png"><br>
<p><img class="tex" alt="\displaystyle f(x) = \frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \hat{f}_2(\omega) e^{i \omega\cdot x} \, d \omega \ " src="wikipedia-Fourier_Transform_pliki/4130cb04b711f02aa75e6b687882c929.png"></p>
</td>
</tr>
<tr>
<th>unitary</th>
<td><img class="tex" alt="\displaystyle \hat{f}_3(\omega) \ \stackrel{\mathrm{def}}{=}\  \frac{1}{(2 \pi)^{n/2}} \int_{\mathbb{R}^n} f(x) \ e^{-i \omega\cdot x}\, dx = \frac{1}{(2 \pi)^{n/2}} \hat{f}_1\left(\frac{\omega}{2 \pi} \right) = \frac{1}{(2 \pi)^{n/2}} \hat{f}_2(\omega) " src="wikipedia-Fourier_Transform_pliki/ec22d9ed6e37c0a87693cbd5b5342172.png"><br>
<p><img class="tex" alt="\displaystyle f(x) = \frac{1}{(2 \pi)^{n/2}} \int_{\mathbb{R}^n} \hat{f}_3(\omega)e^{i \omega\cdot x}\, d \omega \ " src="wikipedia-Fourier_Transform_pliki/dcae8660c6aa2890ff2804857e662812.png"></p>
</td>
</tr>
</tbody></table>
<p>The ordinary-frequency convention (which is used in this article) is the one most often found in the <a href="http://en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematics</a> literature.<sup class="Template-Fact" title="This claim needs references to reliable sources from July 2009" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a></i>]</sup> In the <a href="http://en.wikipedia.org/wiki/Physics" title="Physics">physics</a> literature, the two angular-frequency conventions are more commonly used.<sup class="Template-Fact" title="This claim needs references to reliable sources from July 2009" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a></i>]</sup></p>
<p>As discussed above, the <a href="http://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29" title="Characteristic function (probability theory)">characteristic function</a> of a random variable is the same as the <a href="http://en.wikipedia.org/wiki/Fourier_transform#Fourier.E2.80.93Stieltjes_transform" title="Fourier transform">Fourier–Stieltjes transform</a>
 of its distribution measure, but in this context it is typical to take a
 different convention for the constants. Typically characteristic 
function is defined <img class="tex" alt="E(e^{it\cdot X})=\int e^{it\cdot x}d\mu_X(x)" src="wikipedia-Fourier_Transform_pliki/dc0340ac527b3858bde3abb458129c27.png">. As in the case of the "non-unitary angular frequency" convention above, there is no factor of 2<i>π</i>
 appearing in either of the integral, or in the exponential. Unlike any 
of the conventions appearing above, this convention takes the opposite 
sign in the exponential.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=29" title="Edit section: Tables of important Fourier transforms">edit</a>]</span> <span class="mw-headline" id="Tables_of_important_Fourier_transforms">Tables of important Fourier transforms</span></h2>
<p>The following tables record some closed form Fourier transforms. For functions <i>ƒ</i>(<i>x</i>) , <i>g</i>(<i>x</i>) and <i>h</i>(<i>x</i>) denote their Fourier transforms by <img class="tex" alt="\hat{f}" src="wikipedia-Fourier_Transform_pliki/8205299e2dea57e730eb520073ec705a.png">, <img class="tex" alt="\hat{g}" src="wikipedia-Fourier_Transform_pliki/9d82cd62f6e11a82be1ba79544daf6c0.png">, and <img class="tex" alt="\hat{h}" src="wikipedia-Fourier_Transform_pliki/3099fd2c7357d4fd0e30671c2625b20e.png">
 respectively. Only the three most common conventions are included. It 
may be useful to notice that entry 105 gives a relationship between the 
Fourier transform of a function and the original function, which can be 
seen as relating the Fourier transform and its inverse.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=30" title="Edit section: Functional relationships">edit</a>]</span> <span class="mw-headline" id="Functional_relationships">Functional relationships</span></h3>
<p>The Fourier transforms in this table may be found in (<a href="#CITEREFErd.C3.A9lyi1954">Erdélyi 1954</a>) or the appendix of (<a href="#CITEREFKammler2000">Kammler 2000</a>).</p>
<table class="wikitable">
<tbody><tr>
<th></th>
<th>Function</th>
<th>Fourier transform<br>
unitary, ordinary frequency</th>
<th>Fourier transform<br>
unitary, angular frequency</th>
<th>Fourier transform<br>
non-unitary, angular frequency</th>
<th>Remarks</th>
</tr>
<tr>
<td></td>
<td align="center"><img class="tex" alt="\displaystyle f(x)\," src="wikipedia-Fourier_Transform_pliki/9484432986feef752f2e55e834bcc60c.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \hat{f}(\xi)=" src="wikipedia-Fourier_Transform_pliki/ea8c0df9aacf6882a98b77b7cbda942b.png">
<p><img class="tex" alt="\displaystyle \int_{-\infty}^{\infty}f(x) e^{-2\pi i x\xi}\, dx " src="wikipedia-Fourier_Transform_pliki/529bd431a4ce486e9e67eedcc9535c0d.png"></p>
</td>
<td align="center"><img class="tex" alt="\displaystyle \hat{f}(\omega)=" src="wikipedia-Fourier_Transform_pliki/83abc8c2712e5abe8e9f1a76de164b3e.png"><img class="tex" alt="\displaystyle \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(x) e^{-i \omega x}\, dx " src="wikipedia-Fourier_Transform_pliki/2152e27979f2f501693d8ffcbf19a5b2.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \hat{f}(\nu)=" src="wikipedia-Fourier_Transform_pliki/8bc3770b7c4d16d2f3bb87f1c4b0c4a8.png">
<p><img class="tex" alt="\displaystyle \int_{-\infty}^{\infty}f(x) e^{-i \nu x}\, dx " src="wikipedia-Fourier_Transform_pliki/d0c569ac204695f3c2380bbf9accd7ee.png"></p>
</td>
<td>Definition</td>
</tr>
<tr>
<td>101</td>
<td><img class="tex" alt="\displaystyle a\cdot f(x) + b\cdot g(x)\," src="wikipedia-Fourier_Transform_pliki/d5b6027c079272d5b7bb3007ec217a82.png"></td>
<td><img class="tex" alt="\displaystyle a\cdot \hat{f}(\xi) + b\cdot \hat{g}(\xi)\," src="wikipedia-Fourier_Transform_pliki/deb96bb7728ca2959a5a50433586f6a2.png"></td>
<td><img class="tex" alt="\displaystyle a\cdot \hat{f}(\omega) + b\cdot \hat{g}(\omega)\," src="wikipedia-Fourier_Transform_pliki/34d739736abe2d26582184e10c5d50fe.png"></td>
<td><img class="tex" alt="\displaystyle a\cdot \hat{f}(\nu) + b\cdot \hat{g}(\nu)\," src="wikipedia-Fourier_Transform_pliki/98915e9363041c858818e38905730cd1.png"></td>
<td>Linearity</td>
</tr>
<tr>
<td>102</td>
<td><img class="tex" alt="\displaystyle f(x - a)\," src="wikipedia-Fourier_Transform_pliki/c867a8d3717762af1748877e616b7f47.png"></td>
<td><img class="tex" alt="\displaystyle e^{-2\pi i a \xi} \hat{f}(\xi)\," src="wikipedia-Fourier_Transform_pliki/4062a1dc971e78cd602d6ebe86d0646a.png"></td>
<td><img class="tex" alt="\displaystyle e^{- i a \omega} \hat{f}(\omega)\," src="wikipedia-Fourier_Transform_pliki/45c3708856c61229637d6b8efe66503a.png"></td>
<td><img class="tex" alt="\displaystyle e^{- i a \nu} \hat{f}(\nu)\," src="wikipedia-Fourier_Transform_pliki/7b6f528c90b7139a97abbf2d9da6e43f.png"></td>
<td>Shift in time domain</td>
</tr>
<tr>
<td>103</td>
<td><img class="tex" alt="\displaystyle e^{ 2\pi iax} f(x)\," src="wikipedia-Fourier_Transform_pliki/ae0b8dd4f6f784bb3e1c316d3b7aeb76.png"></td>
<td><img class="tex" alt="\displaystyle \hat{f} \left(\xi - a\right)\," src="wikipedia-Fourier_Transform_pliki/5b8ad1414341d42cc1635c4bd7b61303.png"></td>
<td><img class="tex" alt="\displaystyle \hat{f}(\omega - 2\pi a)\," src="wikipedia-Fourier_Transform_pliki/bf02df3e408c25b254c85295036febee.png"></td>
<td><img class="tex" alt="\displaystyle \hat{f}(\nu - 2\pi a)\," src="wikipedia-Fourier_Transform_pliki/2e91cc1d425acd65641cba8281f4641a.png"></td>
<td>Shift in frequency domain, dual of 102</td>
</tr>
<tr>
<td>104</td>
<td><img class="tex" alt="\displaystyle f(a x)\," src="wikipedia-Fourier_Transform_pliki/a46e997979c8c81212954e0b832e9196.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{|a|} \hat{f}\left( \frac{\xi}{a} \right)\," src="wikipedia-Fourier_Transform_pliki/1a850ceca7f0ce2044da7882b5dcd0d8.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{|a|} \hat{f}\left( \frac{\omega}{a} \right)\," src="wikipedia-Fourier_Transform_pliki/5fbe81b952289680ab68385e37289a97.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{|a|} \hat{f}\left( \frac{\nu}{a} \right)\," src="wikipedia-Fourier_Transform_pliki/1c0d834a44d651eefac02423d470e4e4.png"></td>
<td>Scaling in the time domain. If <img class="tex" alt="\displaystyle |a|\," src="wikipedia-Fourier_Transform_pliki/1a34b48d3bdb3f4d1395355e2bf7881a.png"> is large, then <img class="tex" alt="\displaystyle f(a x)\," src="wikipedia-Fourier_Transform_pliki/a46e997979c8c81212954e0b832e9196.png"> is concentrated around 0 and <img class="tex" alt="\displaystyle \frac{1}{|a|}\hat{f} \left( \frac{\omega}{a} \right)\," src="wikipedia-Fourier_Transform_pliki/5fbe81b952289680ab68385e37289a97.png"> spreads out and flattens.</td>
</tr>
<tr>
<td>105</td>
<td><img class="tex" alt="\displaystyle \hat{f}(x)\," src="wikipedia-Fourier_Transform_pliki/4b3551f0f6199247695ecc4cbdb36330.png"></td>
<td><img class="tex" alt="\displaystyle f(-\xi)\," src="wikipedia-Fourier_Transform_pliki/69a141d180a9d6ba03ce9db4d3ec45cd.png"></td>
<td><img class="tex" alt="\displaystyle f(-\omega)\," src="wikipedia-Fourier_Transform_pliki/ba33c4e0277f6c78686e686eb98d2643.png"></td>
<td><img class="tex" alt="\displaystyle 2\pi f(-\nu)\," src="wikipedia-Fourier_Transform_pliki/614e7ba1cedd259b23e2770abc95bf83.png"></td>
<td>Duality. Here <img class="tex" alt="\hat{f}" src="wikipedia-Fourier_Transform_pliki/8205299e2dea57e730eb520073ec705a.png"> needs to be calculated using the same method as Fourier transform column. Results from swapping "dummy" variables of <img class="tex" alt="\displaystyle x \," src="wikipedia-Fourier_Transform_pliki/a5d5b0773cdc535b7f6cf5b316b70a35.png"> and <img class="tex" alt="\displaystyle \xi \," src="wikipedia-Fourier_Transform_pliki/a1742be8c066f5cdc2e1e428204e11f8.png"> or <img class="tex" alt="\displaystyle \omega \," src="wikipedia-Fourier_Transform_pliki/fab6d99343a9ce19f9d16dcbd1c2d375.png"> or <img class="tex" alt="\displaystyle \nu \," src="wikipedia-Fourier_Transform_pliki/5c54e834d32fdfa9f6954d345de5899c.png">.</td>
</tr>
<tr>
<td>106</td>
<td><img class="tex" alt="\displaystyle \frac{d^n f(x)}{dx^n}\," src="wikipedia-Fourier_Transform_pliki/3048bf7e562e699f5dc4e3de5fea618d.png"></td>
<td><img class="tex" alt="\displaystyle  (2\pi i\xi)^n  \hat{f}(\xi)\," src="wikipedia-Fourier_Transform_pliki/5e88670d05d49450c4669e47209745a5.png"></td>
<td><img class="tex" alt="\displaystyle (i\omega)^n  \hat{f}(\omega)\," src="wikipedia-Fourier_Transform_pliki/6b81834289df1c632a08a0e7def8a0b4.png"></td>
<td><img class="tex" alt="\displaystyle (i\nu)^n  \hat{f}(\nu)\," src="wikipedia-Fourier_Transform_pliki/9878f4f3438dbce63f04c55ff9a2073b.png"></td>
<td></td>
</tr>
<tr>
<td>107</td>
<td><img class="tex" alt="\displaystyle x^n f(x)\," src="wikipedia-Fourier_Transform_pliki/647011377a27cf6566e100a93914c4da.png"></td>
<td><img class="tex" alt="\displaystyle \left (\frac{i}{2\pi}\right)^n \frac{d^n \hat{f}(\xi)}{d\xi^n}\," src="wikipedia-Fourier_Transform_pliki/09630588111231bcdeb3b1dae5c9ef91.png"></td>
<td><img class="tex" alt="\displaystyle i^n \frac{d^n \hat{f}(\omega)}{d\omega^n}" src="wikipedia-Fourier_Transform_pliki/be7cc88e48ffec36a1b94c475485770e.png"></td>
<td><img class="tex" alt="\displaystyle i^n \frac{d^n \hat{f}(\nu)}{d\nu^n}" src="wikipedia-Fourier_Transform_pliki/f1df22c094a2771f637075321ab93183.png"></td>
<td>This is the dual of 106</td>
</tr>
<tr>
<td>108</td>
<td><img class="tex" alt="\displaystyle (f * g)(x)\," src="wikipedia-Fourier_Transform_pliki/85d589ad4be0cfdc5d39ab1dc2b80147.png"></td>
<td><img class="tex" alt="\displaystyle \hat{f}(\xi) \hat{g}(\xi)\," src="wikipedia-Fourier_Transform_pliki/1c974d2e4859400e2800a009c27a520a.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{2\pi} \hat{f}(\omega) \hat{g}(\omega)\," src="wikipedia-Fourier_Transform_pliki/0ab7c57f9b1ec74052bd0c3a21a0d2df.png"></td>
<td><img class="tex" alt="\displaystyle \hat{f}(\nu) \hat{g}(\nu)\," src="wikipedia-Fourier_Transform_pliki/b30b56b43fd1eab7ec950ecf4839b72e.png"></td>
<td>The notation <img class="tex" alt="\displaystyle f * g\," src="wikipedia-Fourier_Transform_pliki/ca6614ae2c105999bcdfec6451288233.png"> denotes the <a href="http://en.wikipedia.org/wiki/Convolution" title="Convolution">convolution</a> of <img class="tex" alt="\displaystyle f\," src="wikipedia-Fourier_Transform_pliki/0b496569f09220481b561698c1926301.png"> and <img class="tex" alt="\displaystyle g\," src="wikipedia-Fourier_Transform_pliki/89e63b2eafb7810b21241e23cb0b3c32.png"> — this rule is the <a href="http://en.wikipedia.org/wiki/Convolution_theorem" title="Convolution theorem">convolution theorem</a></td>
</tr>
<tr>
<td>109</td>
<td><img class="tex" alt="\displaystyle f(x) g(x)\," src="wikipedia-Fourier_Transform_pliki/c04185e822e6f6e1bc91725aaf3be022.png"></td>
<td><img class="tex" alt="\displaystyle (\hat{f} * \hat{g})(\xi)\," src="wikipedia-Fourier_Transform_pliki/54fb5ff580cc616ec28d189cebb51bd0.png"></td>
<td><img class="tex" alt="\displaystyle (\hat{f} * \hat{g})(\omega) \over \sqrt{2\pi}\," src="wikipedia-Fourier_Transform_pliki/9a21e24c700891d60de126ac6af84ab9.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{2\pi}(\hat{f} * \hat{g})(\nu)\," src="wikipedia-Fourier_Transform_pliki/7b3d67389b3cb3fa3cdcaf0ebb8607e5.png"></td>
<td>This is the dual of 108</td>
</tr>
<tr>
<td>110</td>
<td>For <img class="tex" alt="\displaystyle f(x) \," src="wikipedia-Fourier_Transform_pliki/9484432986feef752f2e55e834bcc60c.png"> a purely real</td>
<td><img class="tex" alt="\displaystyle \hat{f}(-\xi) = \overline{\hat{f}(\xi)}\," src="wikipedia-Fourier_Transform_pliki/eaef829aebe8a332ba21ee750bf9620e.png"></td>
<td><img class="tex" alt="\displaystyle \hat{f}(-\omega) = \overline{\hat{f}(\omega)}\," src="wikipedia-Fourier_Transform_pliki/09a3c87da1e469c463f40e8d56bfebbc.png"></td>
<td><img class="tex" alt="\displaystyle \hat{f}(-\nu) = \overline{\hat{f}(\nu)}\," src="wikipedia-Fourier_Transform_pliki/84d1e1b4ddaf033af5a67840cc62734e.png"></td>
<td>Hermitian symmetry. <img class="tex" alt="\displaystyle \overline{z}\," src="wikipedia-Fourier_Transform_pliki/f2c40c9e5fdef1f4fcac79677383b168.png"> indicates the <a href="http://en.wikipedia.org/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a>.</td>
</tr>
<tr>
<td>111</td>
<td>For <img class="tex" alt="\displaystyle f(x) \," src="wikipedia-Fourier_Transform_pliki/9484432986feef752f2e55e834bcc60c.png"> a purely real <a href="http://en.wikipedia.org/wiki/Even_function" title="Even function" class="mw-redirect">even function</a></td>
<td colspan="3" align="center"><img class="tex" alt="\displaystyle \hat{f}(\omega)" src="wikipedia-Fourier_Transform_pliki/045fbfee2cc5db083f307b42821a6fb1.png">, <img class="tex" alt="\displaystyle \hat{f}(\xi)" src="wikipedia-Fourier_Transform_pliki/935e10e9331d95951f35acd9f661bc47.png"> and <img class="tex" alt="\displaystyle \hat{f}(\nu)\," src="wikipedia-Fourier_Transform_pliki/d7ce87c493a2fea4e866b4f1cfcb5fa8.png"> are purely real <a href="http://en.wikipedia.org/wiki/Even_function" title="Even function" class="mw-redirect">even functions</a>.</td>
<td></td>
</tr>
<tr>
<td>112</td>
<td>For <img class="tex" alt="\displaystyle f(x) \," src="wikipedia-Fourier_Transform_pliki/9484432986feef752f2e55e834bcc60c.png"> a purely real <a href="http://en.wikipedia.org/wiki/Odd_function" title="Odd function" class="mw-redirect">odd function</a></td>
<td colspan="3" align="center"><img class="tex" alt="\displaystyle \hat{f}(\omega)" src="wikipedia-Fourier_Transform_pliki/045fbfee2cc5db083f307b42821a6fb1.png">, <img class="tex" alt="\displaystyle \hat{f}(\xi)" src="wikipedia-Fourier_Transform_pliki/935e10e9331d95951f35acd9f661bc47.png"> and <img class="tex" alt="\displaystyle \hat{f}(\nu)" src="wikipedia-Fourier_Transform_pliki/c07e0d626c2179d799eabb52555ba9f8.png"> are purely <a href="http://en.wikipedia.org/wiki/Imaginary_number" title="Imaginary number">imaginary</a> <a href="http://en.wikipedia.org/wiki/Odd_function" title="Odd function" class="mw-redirect">odd functions</a>.</td>
<td></td>
</tr>
</tbody></table>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=31" title="Edit section: Square-integrable functions">edit</a>]</span> <span class="mw-headline" id="Square-integrable_functions">Square-integrable functions</span></h3>
<p>The Fourier transforms in this table may be found in (<a href="#CITEREFCampbellFoster1948">Campbell &amp; Foster 1948</a>), (<a href="#CITEREFErd.C3.A9lyi1954">Erdélyi 1954</a>), or the appendix of (<a href="#CITEREFKammler2000">Kammler 2000</a>).</p>
<table class="wikitable">
<tbody><tr>
<th></th>
<th>Function</th>
<th>Fourier transform<br>
unitary, ordinary frequency</th>
<th>Fourier transform<br>
unitary, angular frequency</th>
<th>Fourier transform<br>
non-unitary, angular frequency</th>
<th>Remarks</th>
</tr>
<tr>
<td></td>
<td align="center"><img class="tex" alt="\displaystyle f(x)" src="wikipedia-Fourier_Transform_pliki/d533bd95f0f8cfec1d9ea4339fae91de.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \hat{f}(\xi)=" src="wikipedia-Fourier_Transform_pliki/ea8c0df9aacf6882a98b77b7cbda942b.png">
<p><img class="tex" alt="\displaystyle \int_{-\infty}^{\infty}f(x) e^{-2\pi ix\xi}\,dx" src="wikipedia-Fourier_Transform_pliki/529bd431a4ce486e9e67eedcc9535c0d.png"></p>
</td>
<td align="center"><img class="tex" alt="\displaystyle \hat{f}(\omega)=" src="wikipedia-Fourier_Transform_pliki/83abc8c2712e5abe8e9f1a76de164b3e.png">
<p><img class="tex" alt="\displaystyle \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(x) e^{-i \omega x}\, dx" src="wikipedia-Fourier_Transform_pliki/2152e27979f2f501693d8ffcbf19a5b2.png"></p>
</td>
<td align="center"><img class="tex" alt=" \hat{f}(\nu)=" src="wikipedia-Fourier_Transform_pliki/f6536490c757708b48c9756e4d449574.png">
<p><img class="tex" alt="\displaystyle \int_{-\infty}^{\infty} f(x) e^{-i\nu x}\, dx" src="wikipedia-Fourier_Transform_pliki/d0c569ac204695f3c2380bbf9accd7ee.png"></p>
</td>
<td></td>
</tr>
<tr>
<td>201</td>
<td><img class="tex" alt="\displaystyle \operatorname{rect}(a x) \," src="wikipedia-Fourier_Transform_pliki/5c30b84426e1f9e637b7a73508c8259a.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{|a|}\cdot \operatorname{sinc}\left(\frac{\xi}{a}\right)" src="wikipedia-Fourier_Transform_pliki/faadfcf7a74d259a04a3d8f6412e8334.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{\sqrt{2 \pi a^2}}\cdot \operatorname{sinc}\left(\frac{\omega}{2\pi a}\right)" src="wikipedia-Fourier_Transform_pliki/6a687135743468c1393a8cc8f7e90f50.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{|a|}\cdot \operatorname{sinc}\left(\frac{\nu}{2\pi a}\right)" src="wikipedia-Fourier_Transform_pliki/2c7cdb84dd40e7f0f04cfececb66eae3.png"></td>
<td>The <a href="http://en.wikipedia.org/wiki/Rectangular_function" title="Rectangular function">rectangular pulse</a> and the <i>normalized</i> <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc function</a>, here defined as sinc(<i>x</i>) = sin(<i>πx</i>)/(<i>πx</i>)</td>
</tr>
<tr>
<td>202</td>
<td><img class="tex" alt="\displaystyle \operatorname{sinc}(a x)\," src="wikipedia-Fourier_Transform_pliki/351634ea12ad93696b9b562dc4a10a52.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{|a|}\cdot \operatorname{rect}\left(\frac{\xi}{a} \right)\," src="wikipedia-Fourier_Transform_pliki/18144cca51aad932406756ae42566cdf.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{\sqrt{2\pi a^2}}\cdot \operatorname{rect}\left(\frac{\omega}{2 \pi a}\right)" src="wikipedia-Fourier_Transform_pliki/cadd0bc5a242d7e59b62cdd9331b7c83.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{|a|}\cdot \operatorname{rect}\left(\frac{\nu}{2 \pi a}\right)" src="wikipedia-Fourier_Transform_pliki/1fc726e340419052c77a925b3d6ec216.png"></td>
<td>Dual of rule 201. The <a href="http://en.wikipedia.org/wiki/Rectangular_function" title="Rectangular function">rectangular function</a> is an ideal <a href="http://en.wikipedia.org/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a>, and the <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc function</a> is the <a href="http://en.wikipedia.org/wiki/Acausal" title="Acausal" class="mw-redirect">non-causal</a> impulse response of such a filter.</td>
</tr>
<tr>
<td>203</td>
<td><img class="tex" alt="\displaystyle \operatorname{sinc}^2 (a x)" src="wikipedia-Fourier_Transform_pliki/87e28dad5849b9a4ce0464e450f0f420.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{|a|}\cdot \operatorname{tri} \left( \frac{\xi}{a} \right) " src="wikipedia-Fourier_Transform_pliki/e8ccecbed51a5e5f5c55db050faae119.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{\sqrt{2\pi a^2}}\cdot \operatorname{tri} \left( \frac{\omega}{2\pi a} \right) " src="wikipedia-Fourier_Transform_pliki/80b6fe2efb23746100eece808c36fd72.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{|a|}\cdot \operatorname{tri} \left( \frac{\nu}{2\pi a} \right) " src="wikipedia-Fourier_Transform_pliki/6932d62b37d7b3661cb13eb80f3c2472.png"></td>
<td>The function tri(<i>x</i>) is the <a href="http://en.wikipedia.org/wiki/Triangular_function" title="Triangular function">triangular function</a></td>
</tr>
<tr>
<td>204</td>
<td><img class="tex" alt="\displaystyle \operatorname{tri} (a x)" src="wikipedia-Fourier_Transform_pliki/bb6bf7639230ccc8cdfc26d66baba3b5.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{|a|}\cdot \operatorname{sinc}^2 \left( \frac{\xi}{a} \right) \," src="wikipedia-Fourier_Transform_pliki/b52ff7ca04ded3ff26d1481e6338c3fd.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{\sqrt{2\pi a^2}} \cdot \operatorname{sinc}^2 \left( \frac{\omega}{2\pi a} \right) " src="wikipedia-Fourier_Transform_pliki/9d2b9c59a2ff99b739ccc52dcc4d5c24.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{|a|} \cdot \operatorname{sinc}^2 \left( \frac{\nu}{2\pi a} \right) " src="wikipedia-Fourier_Transform_pliki/bef11ab9c22f5394b6d35ef81518c78f.png"></td>
<td>Dual of rule 203.</td>
</tr>
<tr>
<td>205</td>
<td><img class="tex" alt="\displaystyle e^{- a x} u(x) \," src="wikipedia-Fourier_Transform_pliki/7586b3bd4134c5ab08ea6a637d4b6b4f.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{a + 2 \pi i \xi}" src="wikipedia-Fourier_Transform_pliki/992478f1db742bd078fdf249ed18cfe4.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{\sqrt{2 \pi} (a + i \omega)}" src="wikipedia-Fourier_Transform_pliki/611a139941905f056f19f562023419ba.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{a + i \nu}" src="wikipedia-Fourier_Transform_pliki/09807e6680ec8e5b0d65f67eb50ad71b.png"></td>
<td>The function <i>u</i>(<i>x</i>) is the <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside unit step function</a> and <i>a</i>&gt;0.</td>
</tr>
<tr>
<td>206</td>
<td><img class="tex" alt="\displaystyle e^{-\alpha x^2}\," src="wikipedia-Fourier_Transform_pliki/25ccecf505bd9bab80260868ba6e10b8.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{\frac{\pi}{\alpha}}\cdot e^{-\frac{(\pi \xi)^2}{\alpha}}" src="wikipedia-Fourier_Transform_pliki/a057d53f2929159f113866f4d36ac8e8.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{\sqrt{2 \alpha}}\cdot e^{-\frac{\omega^2}{4 \alpha}}" src="wikipedia-Fourier_Transform_pliki/f1ab8258d954961afe7dae9fd61b38d6.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{\frac{\pi}{\alpha}}\cdot e^{-\frac{\nu^2}{4 \alpha}}" src="wikipedia-Fourier_Transform_pliki/d754b1cf333489f8e5ef3759f694e1d3.png"></td>
<td>This shows that, for the unitary Fourier transforms, the <a href="http://en.wikipedia.org/wiki/Gaussian_function" title="Gaussian function">Gaussian function</a> exp(−<i>αx</i><sup>2</sup>) is its own Fourier transform for some choice of <i>α</i>. For this to be integrable we must have Re(<i>α</i>)&gt;0.</td>
</tr>
<tr>
<td>207</td>
<td><img class="tex" alt="\displaystyle \operatorname{e}^{-a|x|} \," src="wikipedia-Fourier_Transform_pliki/f43593d2f22e00857ce1ce2a37e1ce79.png"></td>
<td><img class="tex" alt="\displaystyle \frac{2 a}{a^2 + 4 \pi^2 \xi^2} " src="wikipedia-Fourier_Transform_pliki/8eccaacdca9a37aa23b0863a36b9db4e.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{\frac{2}{\pi}} \cdot \frac{a}{a^2 + \omega^2} " src="wikipedia-Fourier_Transform_pliki/b4013ca3038aefc192f451b014a9522c.png"></td>
<td><img class="tex" alt="\displaystyle \frac{2a}{a^2 + \nu^2} " src="wikipedia-Fourier_Transform_pliki/19f8c0b6b5b2b8f082623e73117c7448.png"></td>
<td>For <i>a&gt;0</i>. That is, the Fourier transform of a decaying <a href="http://en.wikipedia.org/wiki/Exponential_function" title="Exponential function">exponential function</a> is a <a href="http://en.wikipedia.org/wiki/Lorentzian_function" title="Lorentzian function" class="mw-redirect">Lorentzian function</a>.</td>
</tr>
<tr>
<td>208</td>
<td><img class="tex" alt="\displaystyle \frac{J_n (x)}{x} \," src="wikipedia-Fourier_Transform_pliki/172f1b3666f2d742ef76d9b342fd7cc5.png"></td>
<td><img class="tex" alt="\displaystyle \frac{2 i}{n} (-i)^n \cdot U_{n-1} (2 \pi \xi)\," src="wikipedia-Fourier_Transform_pliki/4bf7e79692f3784d9e68ea77eac716b5.png"><br>
<p>&nbsp; <img class="tex" alt="\displaystyle \cdot \ \sqrt{1 - 4 \pi^2 \xi^2}  \operatorname{rect}( \pi \xi ) " src="wikipedia-Fourier_Transform_pliki/bcd599d801d9e9904037756448baf889.png"></p>
</td>
<td><img class="tex" alt="\displaystyle \sqrt{\frac{2}{\pi}} \frac{i}{n} (-i)^n \cdot U_{n-1} (\omega)\," src="wikipedia-Fourier_Transform_pliki/c908c5e22a6864f7797af4a06ab3640b.png"><br>
<p>&nbsp; <img class="tex" alt="\displaystyle \cdot \ \sqrt{1 - \omega^2} \operatorname{rect} \left( \frac{\omega}{2} \right) " src="wikipedia-Fourier_Transform_pliki/1a11913bb6517b8504dda4a0f2ba43a4.png"></p>
</td>
<td><img class="tex" alt="\displaystyle \frac{2 i}{n} (-i)^n \cdot U_{n-1} (\nu)\," src="wikipedia-Fourier_Transform_pliki/172a7cb4a61e32f3460768760ce51793.png"><br>
<p>&nbsp; <img class="tex" alt="\displaystyle \cdot \ \sqrt{1 - \nu^2} \operatorname{rect} \left( \frac{\nu}{2} \right) " src="wikipedia-Fourier_Transform_pliki/740a5545ee0073d6df718f2d9407f709.png"></p>
</td>
<td>The functions <i>J<sub>n</sub></i> (<i>x</i>) are the <i>n</i>-th order Bessel functions of the first kind. The functions <i>U<sub>n</sub></i> (<i>x</i>) are the <a href="http://en.wikipedia.org/wiki/Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomial of the second kind</a>. See 315 and 316 below.</td>
</tr>
<tr>
<td>209</td>
<td><img class="tex" alt="\displaystyle \operatorname{sech}(a x) \," src="wikipedia-Fourier_Transform_pliki/a8048eeb2f4e2b4ccba0f0e2d1eb4e95.png"></td>
<td><img class="tex" alt="\displaystyle \frac{\pi}{a} \operatorname{sech} \left( \frac{\pi^2}{ a} \xi \right)" src="wikipedia-Fourier_Transform_pliki/5a435bbef57bc8da3af3347d9c29f18a.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{a}\sqrt{\frac{\pi}{2}}\operatorname{sech}\left( \frac{\pi}{2 a} \omega \right)" src="wikipedia-Fourier_Transform_pliki/4a488a9185e762a6798e1cd47efcddcc.png"></td>
<td><img class="tex" alt="\displaystyle \frac{\pi}{a}\operatorname{sech}\left( \frac{\pi}{2 a} \nu \right)" src="wikipedia-Fourier_Transform_pliki/35a6dd944230f3c2c385a9a36c69fc1e.png"></td>
<td><a href="http://en.wikipedia.org/wiki/Hyperbolic_function" title="Hyperbolic function">Hyperbolic secant</a> is its own Fourier transform</td>
</tr>
</tbody></table>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=32" title="Edit section: Distributions">edit</a>]</span> <span class="mw-headline" id="Distributions">Distributions</span></h3>
<p>The Fourier transforms in this table may be found in (<a href="#CITEREFErd.C3.A9lyi1954">Erdélyi 1954</a>) or the appendix of (<a href="#CITEREFKammler2000">Kammler 2000</a>).</p>
<table class="wikitable">
<tbody><tr>
<th></th>
<th>Function</th>
<th>Fourier transform<br>
unitary, ordinary frequency</th>
<th>Fourier transform<br>
unitary, angular frequency</th>
<th>Fourier transform<br>
non-unitary, angular frequency</th>
<th>Remarks</th>
</tr>
<tr>
<td></td>
<td align="center"><img class="tex" alt="\displaystyle f(x)" src="wikipedia-Fourier_Transform_pliki/d533bd95f0f8cfec1d9ea4339fae91de.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \hat{f}(\xi)=" src="wikipedia-Fourier_Transform_pliki/ea8c0df9aacf6882a98b77b7cbda942b.png">
<p><img class="tex" alt="\displaystyle \int_{-\infty}^{\infty}f(x) e^{-2\pi ix\xi}\,dx" src="wikipedia-Fourier_Transform_pliki/529bd431a4ce486e9e67eedcc9535c0d.png"></p>
</td>
<td align="center"><img class="tex" alt=" \hat{f}(\omega)=" src="wikipedia-Fourier_Transform_pliki/55ef8b1f4770724bf000af3d66dd81c5.png">
<p><img class="tex" alt="\displaystyle \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(x) e^{-i \omega x}\, dx" src="wikipedia-Fourier_Transform_pliki/2152e27979f2f501693d8ffcbf19a5b2.png"></p>
</td>
<td align="center"><img class="tex" alt="\displaystyle \hat{f}(\nu)=" src="wikipedia-Fourier_Transform_pliki/8bc3770b7c4d16d2f3bb87f1c4b0c4a8.png">
<p><img class="tex" alt="\displaystyle \int_{-\infty}^{\infty} f(x) e^{-i\nu x}\, dx" src="wikipedia-Fourier_Transform_pliki/d0c569ac204695f3c2380bbf9accd7ee.png"></p>
</td>
<td></td>
</tr>
<tr>
<td>301</td>
<td><img class="tex" alt="\displaystyle 1" src="wikipedia-Fourier_Transform_pliki/bbc01fffff9b6c6427b386b48552a227.png"></td>
<td><img class="tex" alt="\displaystyle \delta(\xi)" src="wikipedia-Fourier_Transform_pliki/9b44585d642bf153f9ac8df9c9932b03.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{2\pi}\cdot \delta(\omega)" src="wikipedia-Fourier_Transform_pliki/76502e12f04d53c11a2ce89ce50de3dc.png"></td>
<td><img class="tex" alt="\displaystyle 2\pi\delta(\nu)" src="wikipedia-Fourier_Transform_pliki/cd4ebca249ed778766f60b2fcdee9d30.png"></td>
<td>The distribution <i>δ</i>(<i>ξ</i>) denotes the <a href="http://en.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>.</td>
</tr>
<tr>
<td>302</td>
<td><img class="tex" alt="\displaystyle \delta(x)\," src="wikipedia-Fourier_Transform_pliki/70037b476676caebe8bdd98bc12d46ec.png"></td>
<td><img class="tex" alt="\displaystyle 1" src="wikipedia-Fourier_Transform_pliki/bbc01fffff9b6c6427b386b48552a227.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{\sqrt{2\pi}}\," src="wikipedia-Fourier_Transform_pliki/94a1a6c84b19a45aa6417991225a77a8.png"></td>
<td><img class="tex" alt="\displaystyle 1" src="wikipedia-Fourier_Transform_pliki/bbc01fffff9b6c6427b386b48552a227.png"></td>
<td>Dual of rule 301.</td>
</tr>
<tr>
<td>303</td>
<td><img class="tex" alt="\displaystyle e^{i a x}" src="wikipedia-Fourier_Transform_pliki/1a03ae428735720252c40a614ed5557d.png"></td>
<td><img class="tex" alt="\displaystyle \delta\left(\xi - \frac{a}{2\pi}\right)" src="wikipedia-Fourier_Transform_pliki/328f831746f9f086e7eaef71d1fcc16d.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{2 \pi}\cdot \delta(\omega - a)" src="wikipedia-Fourier_Transform_pliki/43e53c7a16e9288b43deb6a08a27a0bd.png"></td>
<td><img class="tex" alt="\displaystyle 2 \pi\delta(\nu - a)" src="wikipedia-Fourier_Transform_pliki/77b692ccf1a4bd2eba42e0a3b78efe6d.png"></td>
<td>This follows from 103 and 301.</td>
</tr>
<tr>
<td>304</td>
<td><img class="tex" alt="\displaystyle \cos (a x)" src="wikipedia-Fourier_Transform_pliki/727cc7542c8173f4a09c4daa3f46748c.png"></td>
<td><img class="tex" alt="\displaystyle \frac{\displaystyle \delta\left(\xi - \frac{a}{2\pi}\right)+\delta\left(\xi+\frac{a}{2\pi}\right)}{2}" src="wikipedia-Fourier_Transform_pliki/a6ea37c221a0beb1c9e71e86369f9b96.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{2 \pi}\cdot\frac{\delta(\omega-a)+\delta(\omega+a)}{2}\," src="wikipedia-Fourier_Transform_pliki/2fa53d4a747d410986a215232d62bc2f.png"></td>
<td><img class="tex" alt="\displaystyle \pi\left(\delta(\nu-a)+\delta(\nu+a)\right)" src="wikipedia-Fourier_Transform_pliki/6f5b06c670d3e001ab05c82f5843539a.png"></td>
<td>This follows from rules 101 and 303 using <a href="http://en.wikipedia.org/wiki/Eulers_formula_in_complex_analysis" title="Eulers formula in complex analysis" class="mw-redirect">Euler's formula</a>: <img class="tex" alt="\displaystyle\cos(a x) = (e^{i a x} + e^{-i a x})/2." src="wikipedia-Fourier_Transform_pliki/4c0976e3c77aa6903f22f4c6f2a11cd9.png"></td>
</tr>
<tr>
<td>305</td>
<td><img class="tex" alt="\displaystyle \sin( ax)" src="wikipedia-Fourier_Transform_pliki/f0275ce4f9b188b6b4a8f3158e598eb8.png"></td>
<td><img class="tex" alt="\displaystyle \frac{\displaystyle\delta\left(\xi-\frac{a}{2\pi}\right)-\delta\left(\xi+\frac{a}{2\pi}\right)}{2i}" src="wikipedia-Fourier_Transform_pliki/cc07778acddfb3e3a51d3d221512dcc8.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{2 \pi}\cdot\frac{\delta(\omega-a)-\delta(\omega+a)}{2i}" src="wikipedia-Fourier_Transform_pliki/e0ef5fd133a26339de26c9cdb9ddf3ee.png"></td>
<td><img class="tex" alt="\displaystyle -i\pi\left(\delta(\nu-a)-\delta(\nu+a)\right)" src="wikipedia-Fourier_Transform_pliki/770a63f9211695f594e57bab2834d5df.png"></td>
<td>This follows from 101 and 303 using <img class="tex" alt="\displaystyle\sin(a x) = (e^{i a x} - e^{-i a x})/(2i)." src="wikipedia-Fourier_Transform_pliki/e200911348a191f7fb958b3d5fdc6b0e.png"></td>
</tr>
<tr>
<td>306</td>
<td><img class="tex" alt="\displaystyle \cos ( a x^2 ) " src="wikipedia-Fourier_Transform_pliki/97c9f87dfe7d6a89ddc5678e81303be3.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{\frac{\pi}{a}} \cos \left( \frac{\pi^2 \xi^2}{a} - \frac{\pi}{4} \right)  " src="wikipedia-Fourier_Transform_pliki/8064adeebc452ce9adcb1c8951dfba4b.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{\sqrt{2 a}} \cos \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right) " src="wikipedia-Fourier_Transform_pliki/952a9c96f5e24fb33eb5d255fcc9736a.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{\frac{\pi}{a}} \cos \left( \frac{\nu^2}{4 a} - \frac{\pi}{4} \right) " src="wikipedia-Fourier_Transform_pliki/96838b381aac1cda39f5dbb6c8fc0679.png"></td>
<td></td>
</tr>
<tr>
<td>307</td>
<td><img class="tex" alt="\displaystyle \sin ( a x^2 ) \," src="wikipedia-Fourier_Transform_pliki/837b0ef264602bea9ef8b577bf136bfa.png"></td>
<td><img class="tex" alt="\displaystyle - \sqrt{\frac{\pi}{a}}  \sin \left( \frac{\pi^2 \xi^2}{a} - \frac{\pi}{4} \right)  " src="wikipedia-Fourier_Transform_pliki/706f34f52c3e9f80972ef65ac1fcc60f.png"></td>
<td><img class="tex" alt="\displaystyle \frac{-1}{\sqrt{2 a}} \sin \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right) " src="wikipedia-Fourier_Transform_pliki/d60ab03810c49bad3b23872f835a7665.png"></td>
<td><img class="tex" alt="\displaystyle -\sqrt{\frac{\pi}{a}}\sin \left( \frac{\nu^2}{4 a} - \frac{\pi}{4} \right)" src="wikipedia-Fourier_Transform_pliki/d82701eec58fa359437c191fba35c218.png"></td>
<td></td>
</tr>
<tr>
<td>308</td>
<td><img class="tex" alt="\displaystyle x^n\," src="wikipedia-Fourier_Transform_pliki/128088361e882a4a60d0f81335d4a7fb.png"></td>
<td><img class="tex" alt="\displaystyle \left(\frac{i}{2\pi}\right)^n \delta^{(n)} (\xi)\," src="wikipedia-Fourier_Transform_pliki/9ec7f9e62ae8fd868113ee0683802c8f.png"></td>
<td><img class="tex" alt="\displaystyle i^n \sqrt{2\pi} \delta^{(n)} (\omega)\," src="wikipedia-Fourier_Transform_pliki/77203aa3618bcfd13f74aa2c9ce4e5de.png"></td>
<td><img class="tex" alt="\displaystyle 2\pi i^n\delta^{(n)} (\nu)\," src="wikipedia-Fourier_Transform_pliki/a50a3df5a0b6fabfd766e51fca150a8d.png"></td>
<td>Here, <i>n</i> is a <a href="http://en.wikipedia.org/wiki/Natural_number" title="Natural number">natural number</a> and <img class="tex" alt="\displaystyle\delta^{(n)}(\xi)" src="wikipedia-Fourier_Transform_pliki/2c98d9d4bf1da7ec52eb0c3acda28040.png"> is the <i>n</i>-th
 distribution derivative of the Dirac delta function. This rule follows 
from rules 107 and 301. Combining this rule with 101, we can transform 
all <a href="http://en.wikipedia.org/wiki/Polynomial" title="Polynomial">polynomials</a>.</td>
</tr>
<tr>
<td>309</td>
<td><img class="tex" alt="\displaystyle \frac{1}{x}" src="wikipedia-Fourier_Transform_pliki/e6d4b9d4a28b941ee9c99d4577220921.png"></td>
<td><img class="tex" alt="\displaystyle -i\pi\sgn(\xi)" src="wikipedia-Fourier_Transform_pliki/cd23f252834a0b88f9d5c4c2a0936859.png"></td>
<td><img class="tex" alt="\displaystyle -i\sqrt{\frac{\pi}{2}}\sgn(\omega)" src="wikipedia-Fourier_Transform_pliki/8c9e7c778bcb08d7a7eafa2be1705597.png"></td>
<td><img class="tex" alt="\displaystyle -i\pi\sgn(\nu)" src="wikipedia-Fourier_Transform_pliki/3a80eb2b6b5920da5e5ebd709fdc0ad9.png"></td>
<td>Here sgn(<i>ξ</i>) is the <a href="http://en.wikipedia.org/wiki/Sign_function" title="Sign function">sign function</a>. Note that 1/<i>x</i> is not a distribution. It is necessary to use the <a href="http://en.wikipedia.org/wiki/Cauchy_principal_value" title="Cauchy principal value">Cauchy principal value</a> when testing against Schwartz functions. This rule is useful in studying the <a href="http://en.wikipedia.org/wiki/Hilbert_transform" title="Hilbert transform">Hilbert transform</a>.</td>
</tr>
<tr>
<td>310</td>
<td><img class="tex" alt="\displaystyle \frac{1}{x^n}&nbsp;:= \frac{(-1)^{n-1}}{(n-1)!}\frac{d^n}{dx^n}\log |x|" src="wikipedia-Fourier_Transform_pliki/551adb2c6920cd288412cb0130039eed.png"></td>
<td><img class="tex" alt="\displaystyle -i\pi \frac{(-2\pi i\xi)^{n-1}}{(n-1)!} \sgn(\xi)" src="wikipedia-Fourier_Transform_pliki/8be6d6529fd10aff93946a3370629d7e.png"></td>
<td><img class="tex" alt="\displaystyle -i\sqrt{\frac{\pi}{2}}\cdot \frac{(-i\omega)^{n-1}}{(n-1)!}\sgn(\omega)" src="wikipedia-Fourier_Transform_pliki/6e9fdc9f036b2a43448f5608e50383bc.png"></td>
<td><img class="tex" alt="\displaystyle -i\pi \frac{(-i\nu)^{n-1}}{(n-1)!}\sgn(\nu)" src="wikipedia-Fourier_Transform_pliki/25241839167a54c157dc7835010664c1.png"></td>
<td>1/<i>x</i><sup><i>n</i></sup> is the <a href="http://en.wikipedia.org/wiki/Homogeneous_distribution" title="Homogeneous distribution">homogeneous distribution</a> defined by the distributional derivative <img class="tex" alt="\textstyle\frac{(-1)^{n-1}}{(n-1)!}\frac{d^n}{dx^n}\log|x|" src="wikipedia-Fourier_Transform_pliki/8fa0bb25572f1b2c20c2fe30492bed17.png"></td>
</tr>
<tr>
<td>311</td>
<td><img class="tex" alt="\displaystyle |x|^\alpha\," src="wikipedia-Fourier_Transform_pliki/cdd982712c32fa111b9f004fe9cba5b3.png"></td>
<td><img class="tex" alt="\displaystyle -2 \frac{\sin(\pi\alpha/2)\Gamma(\alpha+1)}{|2\pi\xi|^{\alpha+1}}" src="wikipedia-Fourier_Transform_pliki/bbfe98d8a3518f84255c878bc24388ce.png"></td>
<td><img class="tex" alt="\displaystyle \frac{-2}{\sqrt{2\pi}}\frac{\sin(\pi\alpha/2)\Gamma(\alpha+1)}{|\omega|^{\alpha+1}} " src="wikipedia-Fourier_Transform_pliki/62d349d489c928181e72e188b4f8869a.png"></td>
<td><img class="tex" alt="\displaystyle -2\frac{\sin(\pi\alpha/2)\Gamma(\alpha+1)}{|\nu|^{\alpha+1}} " src="wikipedia-Fourier_Transform_pliki/ab627ee7685a4248e779c222674ea410.png"></td>
<td>If Re α &gt; −1, then <span class="texhtml">| <i>x</i> | <sup>α</sup></span> is a locally integrable function, and so a tempered distribution. The function <img class="tex" alt="\alpha\mapsto |x|^\alpha" src="wikipedia-Fourier_Transform_pliki/10aea631bceab4e45bc87a606fd9bd41.png">
 is a holomorphic function from the right half-plane to the space of 
tempered distributions. It admits a unique meromorphic extension to a 
tempered distribution, also denoted <span class="texhtml">| <i>x</i> | <sup>α</sup></span> for α ≠ −2, −4,&nbsp;... (See <a href="http://en.wikipedia.org/wiki/Homogeneous_distribution" title="Homogeneous distribution">homogeneous distribution</a>.)</td>
</tr>
<tr>
<td>312</td>
<td><img class="tex" alt="\displaystyle \sgn(x)" src="wikipedia-Fourier_Transform_pliki/883927c52b316a1276bd4dccb707cff0.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{i\pi \xi}" src="wikipedia-Fourier_Transform_pliki/793122398bd0bbe1c2c142f4b3ee37c4.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{\frac{2}{\pi}}\cdot \frac{1}{i\omega }\," src="wikipedia-Fourier_Transform_pliki/d9ffc56b3ac51c307f93ad09efcfe2e7.png"></td>
<td><img class="tex" alt="\displaystyle \frac{2}{i\nu }" src="wikipedia-Fourier_Transform_pliki/76bfd320ab77de3553739bb3ee9595a4.png"></td>
<td>The dual of rule 309. This time the Fourier transforms need to be considered as <a href="http://en.wikipedia.org/wiki/Cauchy_principal_value" title="Cauchy principal value">Cauchy principal value</a>.</td>
</tr>
<tr>
<td>313</td>
<td><img class="tex" alt="\displaystyle u(x)" src="wikipedia-Fourier_Transform_pliki/29a068f3ebe093b6492b2adae8e380e2.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{2}\left(\frac{1}{i \pi \xi} + \delta(\xi)\right)" src="wikipedia-Fourier_Transform_pliki/b9f1ebdeccd7906a24b53b95244bfdb9.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{\frac{\pi}{2}} \left( \frac{1}{i \pi \omega} + \delta(\omega)\right)" src="wikipedia-Fourier_Transform_pliki/58b2e4fa948d9110fe9e92b8c9dd1d07.png"></td>
<td><img class="tex" alt="\displaystyle \pi\left( \frac{1}{i \pi \nu} + \delta(\nu)\right)" src="wikipedia-Fourier_Transform_pliki/eac68e4b44f2fd1e1a3f1f23030633d1.png"></td>
<td>The function <i>u</i>(<i>x</i>) is the Heaviside <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">unit step function</a>; this follows from rules 101, 301, and 312.</td>
</tr>
<tr>
<td>314</td>
<td><img class="tex" alt="\displaystyle \sum_{n=-\infty}^{\infty} \delta (x - n T)" src="wikipedia-Fourier_Transform_pliki/65f7c225877ba839eddbfc5ef50d10c8.png"></td>
<td><img class="tex" alt="\displaystyle \frac{1}{T} \sum_{k=-\infty}^{\infty} \delta \left( \xi -\frac{k }{T}\right)" src="wikipedia-Fourier_Transform_pliki/d55129171b2d4552386361d787d9be36.png"></td>
<td><img class="tex" alt="\displaystyle \frac{\sqrt{2\pi }}{T}\sum_{k=-\infty}^{\infty} \delta \left( \omega -\frac{2\pi k}{T}\right)" src="wikipedia-Fourier_Transform_pliki/a3f78ed81d157ecbd3f8723f29d102cc.png"></td>
<td><img class="tex" alt="\displaystyle \frac{2\pi}{T}\sum_{k=-\infty}^{\infty} \delta \left( \nu -\frac{2\pi k}{T}\right)" src="wikipedia-Fourier_Transform_pliki/3df3486eb48d48a3b6708bd81e9f9478.png"></td>
<td>This function is known as the <a href="http://en.wikipedia.org/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a> function. This result can be derived from 302 and 102, together with the fact that <img class="tex" alt="\sum_{n=-\infty}^{\infty} e^{inx}=2\pi\sum_{k=-\infty}^{\infty} \delta(x+2\pi k)" src="wikipedia-Fourier_Transform_pliki/a70e9899857c4d7c9dde291b046207f8.png"> as distributions.</td>
</tr>
<tr>
<td>315</td>
<td><img class="tex" alt="\displaystyle J_0 (x)" src="wikipedia-Fourier_Transform_pliki/cb0f12d9179c4bd3dbc40e97b9bbb2c2.png"></td>
<td><img class="tex" alt="\displaystyle \frac{2\, \operatorname{rect}(\pi\xi)}{\sqrt{1 - 4 \pi^2 \xi^2}} " src="wikipedia-Fourier_Transform_pliki/31b44f6638538cb45235b89332ca4e78.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{\frac{2}{\pi}} \cdot \frac{\operatorname{rect}\left( \displaystyle \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} " src="wikipedia-Fourier_Transform_pliki/e9f7c4cd7c5437f9a52eff479c1f2e95.png"></td>
<td><img class="tex" alt="\displaystyle \frac{2\,\operatorname{rect}\left(\displaystyle\frac{\nu}{2} \right)}{\sqrt{1 - \nu^2}}" src="wikipedia-Fourier_Transform_pliki/41976317a6021715a3cd80739a3e05fe.png"></td>
<td>The function <i>J</i><sub>0</sub>(<i>x</i>) is the zeroth order <a href="http://en.wikipedia.org/wiki/Bessel_function" title="Bessel function">Bessel function</a> of first kind.</td>
</tr>
<tr>
<td>316</td>
<td><img class="tex" alt="\displaystyle J_n (x)" src="wikipedia-Fourier_Transform_pliki/d49a18b29c6c754b679cb9d3d8d2269a.png"></td>
<td><img class="tex" alt="\displaystyle \frac{2 (-i)^n T_n (2 \pi \xi) \operatorname{rect}(\pi \xi)}{\sqrt{1 - 4 \pi^2 \xi^2}} " src="wikipedia-Fourier_Transform_pliki/ff8dbef2dda6fa47f911b2ec54ba0b32.png"></td>
<td><img class="tex" alt="\displaystyle \sqrt{\frac{2}{\pi}} \frac{ (-i)^n T_n (\omega) \operatorname{rect} \left( \displaystyle\frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} " src="wikipedia-Fourier_Transform_pliki/9695a3abafb53c41dbd61e73d09d2b65.png"></td>
<td><img class="tex" alt="\displaystyle \frac{2(-i)^n T_n (\nu) \operatorname{rect} \left(\displaystyle \frac{\nu}{2} \right)}{\sqrt{1 - \nu^2}} " src="wikipedia-Fourier_Transform_pliki/0a3a9f118204ea217da810445585be35.png"></td>
<td>This is a generalization of 315. The function <i>J<sub>n</sub></i>(<i>x</i>) is the <i>n</i>-th order <a href="http://en.wikipedia.org/wiki/Bessel_function" title="Bessel function">Bessel function</a> of first kind. The function <i>T<sub>n</sub></i>(<i>x</i>) is the <a href="http://en.wikipedia.org/wiki/Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomial of the first kind</a>.</td>
</tr>
</tbody></table>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=33" title="Edit section: Two-dimensional functions">edit</a>]</span> <span class="mw-headline" id="Two-dimensional_functions">Two-dimensional functions</span></h3>
<table class="wikitable">
<tbody><tr>
<th>Functions (400 to 402)</th>
<th>Fourier transform<br>
unitary, ordinary frequency</th>
<th>Fourier transform<br>
unitary, angular frequency</th>
<th>Fourier transform<br>
non-unitary, angular frequency</th>
</tr>
<tr>
<td align="center"><img class="tex" alt="\displaystyle f(x,y)" src="wikipedia-Fourier_Transform_pliki/b8aeaa721511472d40ab24447aa4fc63.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \hat{f}(\xi_x, \xi_y)=" src="wikipedia-Fourier_Transform_pliki/33024b638719b349d5f1e583f9892d1c.png">
<p><img class="tex" alt="\displaystyle \iint f(x,y) e^{-2\pi i(\xi_x x+\xi_y y)}\,dxdy" src="wikipedia-Fourier_Transform_pliki/0ccb1cabe3ecb10a9dca1a84bb5b16c2.png"></p>
</td>
<td align="center"><img class="tex" alt="\displaystyle \hat{f}(\omega_x,\omega_y)=" src="wikipedia-Fourier_Transform_pliki/28d14bfd4472d8cddd6107472ecd4d4e.png">
<p><img class="tex" alt="\displaystyle \frac{1}{2 \pi} \iint f(x,y) e^{-i (\omega_x x +\omega_y y)}\, dxdy" src="wikipedia-Fourier_Transform_pliki/e89f9164082eba7631429860c8a1738f.png"></p>
</td>
<td align="center"><img class="tex" alt=" \hat{f}(\nu_x,\nu_y)=" src="wikipedia-Fourier_Transform_pliki/c23fdd195c06d794b5cb34f53ad046ee.png">
<p><img class="tex" alt="\displaystyle \iint f(x,y) e^{-i(\nu_x x+\nu_y y)}\, dxdy" src="wikipedia-Fourier_Transform_pliki/9f99ff291c7db6e45cb6454b9905f7a6.png"></p>
</td>
</tr>
<tr>
<td align="center"><img class="tex" alt="\displaystyle e^{-\pi\left(a^2x^2+b^2y^2\right)}" src="wikipedia-Fourier_Transform_pliki/b8765c4480eeb9327af4d6ee56489526.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \frac{1}{|ab|} e^{-\pi\left(\xi_x^2/a^2 + \xi_y^2/b^2\right)}" src="wikipedia-Fourier_Transform_pliki/adcfc565f15e147d12622216a440a869.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \frac{1}{2\pi\cdot|ab|} e^{\frac{-\left(\omega_x^2/a^2 + \omega_y^2/b^2\right)}{4\pi}}" src="wikipedia-Fourier_Transform_pliki/9536c49908b54fbf75e4730000f9acd5.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \frac{1}{|ab|} e^{\frac{-\left(\nu_x^2/a^2 + \nu_y^2/b^2\right)}{4\pi}}" src="wikipedia-Fourier_Transform_pliki/f61eff0b4a116ced303d12447bfa10fa.png"></td>
</tr>
<tr>
<td><img class="tex" alt="\displaystyle \mathrm{circ}(\sqrt{x^2+y^2})" src="wikipedia-Fourier_Transform_pliki/685bb1de39bf5194d17aa5860ca258d4.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \frac{J_1\left(2 \pi \sqrt{\xi_x^2+\xi_y^2}\right)}{\sqrt{\xi_x^2+\xi_y^2}}" src="wikipedia-Fourier_Transform_pliki/35ef315391ece494d627f04585357d23.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \frac{J_1\left(\sqrt{\omega_x^2+\omega_y^2}\right)}{\sqrt{\omega_x^2+\omega_y^2}}" src="wikipedia-Fourier_Transform_pliki/9a15b613c5edfaef5b12607a4df9dcfe.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \frac{2\pi J_1\left(\sqrt{\nu_x^2+\nu_y^2}\right)}{\sqrt{\nu_x^2+\nu_y^2}}" src="wikipedia-Fourier_Transform_pliki/ade543e76541dc87438d36e3981d4cee.png"></td>
</tr>
</tbody></table>
<dl>
<dt>Remarks</dt>
</dl>
<p><i>To 400:</i> The variables <i>ξ<sub>x</sub></i>, <i>ξ<sub>y</sub></i>, <i>ω<sub>x</sub></i>, <i>ω<sub>y</sub></i>, <i>ν<sub>x</sub></i> and <i>ν<sub>y</sub></i> are real numbers. The integrals are taken over the entire plane.</p>
<p><i>To 401:</i> Both functions are Gaussians, which may not have unit volume.</p>
<p><i>To 402:</i> The function is defined by circ(<i>r</i>)=1 0≤<i>r</i>≤1, and is 0 otherwise. This is the Airy distribution, and is expressed using J<sub>1</sub> (the order 1 <a href="http://en.wikipedia.org/wiki/Bessel_function" title="Bessel function">Bessel function</a> of the first kind). (<a href="#CITEREFSteinWeiss1971">Stein &amp; Weiss 1971</a>, Thm. IV.3.3)</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=34" title="Edit section: Formulas for general n-dimensional functions">edit</a>]</span> <span class="mw-headline" id="Formulas_for_general_n-dimensional_functions">Formulas for general <i>n</i>-dimensional functions</span></h3>
<table class="wikitable">
<tbody><tr>
<th></th>
<th>Function</th>
<th>Fourier transform<br>
unitary, ordinary frequency</th>
<th>Fourier transform<br>
unitary, angular frequency</th>
<th>Fourier transform<br>
non-unitary, angular frequency</th>
</tr>
<tr>
<td>500</td>
<td align="center"><img class="tex" alt="\displaystyle f(x)\," src="wikipedia-Fourier_Transform_pliki/9484432986feef752f2e55e834bcc60c.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \hat{f}(\xi)=" src="wikipedia-Fourier_Transform_pliki/ea8c0df9aacf6882a98b77b7cbda942b.png">
<p><img class="tex" alt="\displaystyle \int_{\mathbb{R}^n}f(x) e^{-2\pi i x\cdot\xi }\, d^n x " src="wikipedia-Fourier_Transform_pliki/a1b2b857e6af765633781b2081ffa957.png"></p>
</td>
<td align="center"><img class="tex" alt="\displaystyle \hat{f}(\omega)=" src="wikipedia-Fourier_Transform_pliki/83abc8c2712e5abe8e9f1a76de164b3e.png"><img class="tex" alt="\displaystyle \frac{1}{{(2 \pi)}^{(n/2)}} \int_{\mathbb{R}^n} f(x) e^{-i \omega\cdot x}\, d^nx " src="wikipedia-Fourier_Transform_pliki/3a6f18e751767fe9e36834d2931f6508.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \hat{f}(\nu)=" src="wikipedia-Fourier_Transform_pliki/8bc3770b7c4d16d2f3bb87f1c4b0c4a8.png">
<p><img class="tex" alt="\displaystyle \int_{\mathbb{R}^n}f(x) e^{-i x\cdot\nu }\, d^nx " src="wikipedia-Fourier_Transform_pliki/9c848cd15996ae200e36fc047b46b27e.png"></p>
</td>
</tr>
<tr>
<td>501</td>
<td><img class="tex" alt="\displaystyle \chi_{[0,1]}(|x|)(1-|x|^2)^\delta" src="wikipedia-Fourier_Transform_pliki/4fd9edffd8edd25156a82f85112d142f.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \pi^{-\delta}\Gamma(\delta+1)|\xi|^{-(n/2)-\delta}" src="wikipedia-Fourier_Transform_pliki/c33e2da980a01f1b9cdf41c29fada53c.png"><br>
<img class="tex" alt="\displaystyle \cdot J_{n/2+\delta}(2\pi|\xi|)" src="wikipedia-Fourier_Transform_pliki/a99ab2cbb27b8e1ab7e3b6ba0d175db9.png"></td>
<td align="center"><img class="tex" alt="\displaystyle 2^{-\delta}\Gamma(\delta+1)\left|\omega\right|^{-(n/2)-\delta}" src="wikipedia-Fourier_Transform_pliki/bec204c4fec9f5f9b78a58e589fbcff4.png"><br>
<img class="tex" alt="\displaystyle \cdot J_{n/2+\delta}(|\omega|)" src="wikipedia-Fourier_Transform_pliki/b8cbd1e9dd23f86d7bf4ce4d839a0e1c.png"></td>
<td align="center"><img class="tex" alt="\displaystyle \pi^{-\delta}\Gamma(\delta+1)\left|\frac{\nu}{2\pi}\right|^{-(n/2)-\delta}" src="wikipedia-Fourier_Transform_pliki/6b4b116401acb1219ed557420c4d3e78.png"><br>
<img class="tex" alt="\displaystyle \cdot J_{n/2+\delta}(|\nu|)" src="wikipedia-Fourier_Transform_pliki/31e17b6b389c9366d5e12c1b08051534.png"></td>
</tr>
<tr>
<td>502</td>
<td><img class="tex" alt="\displaystyle |x|^{-\alpha},\quad 0&lt;\operatorname{Re}\, \alpha&lt;n." src="wikipedia-Fourier_Transform_pliki/d60aeb0ad6e0966dbf115f498e90f0f0.png"></td>
<td align="center"><img class="tex" alt="\displaystyle c_\alpha |\xi|^{-(n-\alpha)}" src="wikipedia-Fourier_Transform_pliki/c590f7ca0d2f67d0eb75fb7a8c71a2cc.png"></td>
<td align="center"></td>
<td align="center"></td>
</tr>
</tbody></table>
<dl>
<dt>Remarks</dt>
</dl>
<p><i>To 501</i>: The function χ<sub>[0,1]</sub> is the <a href="http://en.wikipedia.org/wiki/Indicator_function" title="Indicator function">indicator function</a> of the interval [0,&nbsp;1]. The function <i>Γ</i>(<i>x</i>) is the gamma function. The function <i>J</i><sub><i>n</i>/2&nbsp;+&nbsp;<i>δ</i></sub> is a Bessel function of the first kind, with order <i>n</i>/2&nbsp;+&nbsp;<i>δ</i>. Taking <i>n</i>&nbsp;=&nbsp;2 and <i>δ</i>&nbsp;=&nbsp;0 produces 402. (<a href="#CITEREFSteinWeiss1971">Stein &amp; Weiss 1971</a>, Thm. 4.13)</p>
<p><i>To 502</i>: See <a href="http://en.wikipedia.org/wiki/Riesz_potential" title="Riesz potential">Riesz potential</a>. The formula also holds for all α&nbsp;≠&nbsp;−<i>n</i>, −<i>n</i>&nbsp;−&nbsp;1,&nbsp;...
 by analytic continuation, but then the function and its Fourier 
transforms need to be understood as suitably regularized tempered 
distributions. See <a href="http://en.wikipedia.org/wiki/Homogeneous_distribution" title="Homogeneous distribution">homogeneous distribution</a>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=35" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<div style="-moz-column-count:2; column-count:2;">
<ul>
<li><a href="http://en.wikipedia.org/wiki/Fourier_series" title="Fourier series">Fourier series</a></li>
<li><a href="http://en.wikipedia.org/wiki/Fast_Fourier_transform" title="Fast Fourier transform">Fast Fourier transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a>
<ul>
<li><a href="http://en.wikipedia.org/wiki/DFT_matrix" title="DFT matrix">DFT matrix</a></li>
</ul>
</li>
<li><a href="http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Fourier%E2%80%93Deligne_transform" title="Fourier–Deligne transform">Fourier–Deligne transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Fractional_Fourier_transform" title="Fractional Fourier transform">Fractional Fourier transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Linear_canonical_transform" title="Linear canonical transform" class="mw-redirect">Linear canonical transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Fourier_sine_transform" title="Fourier sine transform" class="mw-redirect">Fourier sine transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Space-time_Fourier_transform" title="Space-time Fourier transform">Space-time Fourier transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Short-time_Fourier_transform" title="Short-time Fourier transform">Short-time Fourier transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Fourier_inversion_theorem" title="Fourier inversion theorem">Fourier inversion theorem</a></li>
<li><a href="http://en.wikipedia.org/wiki/Analog_signal_processing" title="Analog signal processing">Analog signal processing</a></li>
<li><a href="http://en.wikipedia.org/wiki/Transform_%28mathematics%29" title="Transform (mathematics)" class="mw-redirect">Transform (mathematics)</a></li>
<li><a href="http://en.wikipedia.org/wiki/Integral_transform" title="Integral transform">Integral transform</a>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Hartley_transform" title="Hartley transform">Hartley transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Hankel_transform" title="Hankel transform">Hankel transform</a></li>
</ul>
</li>
</ul>
</div>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Symbolic_integration" title="Symbolic integration">Symbolic integration</a></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=36" title="Edit section: References">edit</a>]</span> <span class="mw-headline" id="References">References</span></h2>
<ul>
<li><span class="citation" id="CITEREFBoashash2003">Boashash, B., ed. (2003), <i>Time-Frequency Signal Analysis and Processing: A Comprehensive Reference</i>, Oxford: Elsevier Science, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0080443354" title="Special:BookSources/0080443354">0080443354</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Time-Frequency+Signal+Analysis+and+Processing%3A+A+Comprehensive+Reference&amp;rft.date=2003&amp;rft.place=Oxford&amp;rft.pub=Elsevier+Science&amp;rft.isbn=0080443354&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation" id="CITEREFBochner_S..2C_Chandrasekharan_K.1949"><a href="http://en.wikipedia.org/wiki/Salomon_Bochner" title="Salomon Bochner">Bochner S.</a>, <a href="http://en.wikipedia.org/wiki/K._S._Chandrasekharan" title="K. S. Chandrasekharan">Chandrasekharan K.</a> (1949), <i>Fourier Transforms</i>, Princeton University Press</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Fourier+Transforms&amp;rft.aulast=%5B%5BSalomon+Bochner%7CBochner+S.%5D%5D%2C+%5B%5BK.+S.+Chandrasekharan%7CChandrasekharan+K.%5D%5D&amp;rft.au=%5B%5BSalomon+Bochner%7CBochner+S.%5D%5D%2C+%5B%5BK.+S.+Chandrasekharan%7CChandrasekharan+K.%5D%5D&amp;rft.date=1949&amp;rft.pub=Princeton+University+Press&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation" id="CITEREFBracewell2000">Bracewell, R. N. (2000), <i>The Fourier Transform and Its Applications</i> (3rd ed.), Boston: McGraw-Hill, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0071160434" title="Special:BookSources/0071160434">0071160434</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+Fourier+Transform+and+Its+Applications&amp;rft.aulast=Bracewell&amp;rft.aufirst=R.+N.&amp;rft.au=Bracewell%2C%26%2332%3BR.+N.&amp;rft.date=2000&amp;rft.edition=3rd&amp;rft.place=Boston&amp;rft.pub=McGraw-Hill&amp;rft.isbn=0071160434&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFCampbellFoster1948">Campbell, George; Foster, Ronald (1948), <i>Fourier Integrals for Practical Applications</i>, New York: D. Van Nostrand Company, Inc.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Fourier+Integrals+for+Practical+Applications&amp;rft.aulast=Campbell&amp;rft.aufirst=George&amp;rft.au=Campbell%2C%26%2332%3BGeorge&amp;rft.au=Foster%2C%26%2332%3BRonald&amp;rft.date=1948&amp;rft.place=New+York&amp;rft.pub=D.+Van+Nostrand+Company%2C+Inc.&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFDuoandikoetxea2001">Duoandikoetxea, Javier (2001), <i>Fourier Analysis</i>, American Mathematical Society, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-8218-2172-5" title="Special:BookSources/0-8218-2172-5">0-8218-2172-5</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Fourier+Analysis&amp;rft.aulast=Duoandikoetxea&amp;rft.aufirst=Javier&amp;rft.au=Duoandikoetxea%2C%26%2332%3BJavier&amp;rft.date=2001&amp;rft.pub=American+Mathematical+Society&amp;rft.isbn=0-8218-2172-5&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFDymMcKean1985"><a href="http://en.wikipedia.org/wiki/Harry_Dym" title="Harry Dym">Dym, H</a>; McKean, H (1985), <i>Fourier Series and Integrals</i>, Academic Press, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0122264511" title="Special:BookSources/978-0122264511">978-0122264511</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Fourier+Series+and+Integrals&amp;rft.aulast=Dym&amp;rft.aufirst=H&amp;rft.au=Dym%2C%26%2332%3BH&amp;rft.au=McKean%2C%26%2332%3BH&amp;rft.date=1985&amp;rft.pub=Academic+Press&amp;rft.isbn=978-0122264511&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFErd.C3.A9lyi1954">Erdélyi, Arthur, ed. (1954), <i>Tables of Integral Transforms</i>, <b>1</b>, New Your: McGraw-Hill</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Tables+of+Integral+Transforms&amp;rft.date=1954&amp;rft.volume=1&amp;rft.place=New+Your&amp;rft.pub=McGraw-Hill&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation" id="CITEREFFourier1822"><a href="http://en.wikipedia.org/wiki/Joseph_Fourier" title="Joseph Fourier">Fourier, J. B. Joseph</a> (1822), <a href="http://books.google.com/?id=TDQJAAAAIAAJ&amp;printsec=frontcover&amp;dq=Th%C3%A9orie+analytique+de+la+chaleur&amp;q" class="external text" rel="nofollow"><i>Théorie Analytique de la Chaleur</i></a>, Paris<span class="printonly">, <a href="http://books.google.com/?id=TDQJAAAAIAAJ&amp;printsec=frontcover&amp;dq=Th%C3%A9orie+analytique+de+la+chaleur&amp;q" class="external free" rel="nofollow">http://books.google.com/?id=TDQJAAAAIAAJ&amp;printsec=frontcover&amp;dq=Th%C3%A9orie+analytique+de+la+chaleur&amp;q</a></span></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Th%C3%A9orie+Analytique+de+la+Chaleur&amp;rft.aulast=Fourier&amp;rft.aufirst=J.+B.+Joseph&amp;rft.au=Fourier%2C%26%2332%3BJ.+B.+Joseph&amp;rft.date=1822&amp;rft.place=Paris&amp;rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3DTDQJAAAAIAAJ%26printsec%3Dfrontcover%26dq%3DTh%25C3%25A9orie%2Banalytique%2Bde%2Bla%2Bchaleur%26q&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation" id="CITEREFGrafakos2004">Grafakos, Loukas (2004), <i>Classical and Modern Fourier Analysis</i>, Prentice-Hall, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-13-035399-X" title="Special:BookSources/0-13-035399-X">0-13-035399-X</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Classical+and+Modern+Fourier+Analysis&amp;rft.aulast=Grafakos&amp;rft.aufirst=Loukas&amp;rft.au=Grafakos%2C%26%2332%3BLoukas&amp;rft.date=2004&amp;rft.pub=Prentice-Hall&amp;rft.isbn=0-13-035399-X&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFHewittRoss1970">Hewitt, Edwin; Ross, Kenneth A. (1970), <i>Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups</i>, Die Grundlehren der mathematischen Wissenschaften, Band 152, Berlin, New York: <a href="http://en.wikipedia.org/wiki/Springer-Verlag" title="Springer-Verlag" class="mw-redirect">Springer-Verlag</a>, <a href="http://en.wikipedia.org/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a><a href="http://www.ams.org/mathscinet-getitem?mr=0262773" class="external text" rel="nofollow">0262773</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Abstract+harmonic+analysis.+Vol.+II%3A+Structure+and+analysis+for+compact+groups.+Analysis+on+locally+compact+Abelian+groups&amp;rft.aulast=Hewitt&amp;rft.aufirst=Edwin&amp;rft.au=Hewitt%2C%26%2332%3BEdwin&amp;rft.au=Ross%2C%26%2332%3BKenneth+A.&amp;rft.date=1970&amp;rft.series=Die+Grundlehren+der+mathematischen+Wissenschaften%2C+Band+152&amp;rft.place=Berlin%2C+New+York&amp;rft.pub=%5B%5BSpringer-Verlag%5D%5D&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFH.C3.B6rmander1976"><a href="http://en.wikipedia.org/wiki/Lars_H%C3%B6rmander" title="Lars Hörmander">Hörmander, L.</a> (1976), <i>Linear Partial Differential Operators, Volume 1</i>, Springer-Verlag, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/978-3540006626" title="Special:BookSources/978-3540006626">978-3540006626</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Linear+Partial+Differential+Operators%2C+Volume+1&amp;rft.aulast=H%C3%B6rmander&amp;rft.aufirst=L.&amp;rft.au=H%C3%B6rmander%2C%26%2332%3BL.&amp;rft.date=1976&amp;rft.pub=Springer-Verlag&amp;rft.isbn=978-3540006626&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFJames2002">James, J.F. (2002), <i>A Student's Guide to Fourier Transforms</i> (2nd ed.), New York: Cambridge University Press, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-521-00428-4" title="Special:BookSources/0-521-00428-4">0-521-00428-4</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=A+Student%27s+Guide+to+Fourier+Transforms&amp;rft.aulast=James&amp;rft.aufirst=J.F.&amp;rft.au=James%2C%26%2332%3BJ.F.&amp;rft.date=2002&amp;rft.edition=2nd&amp;rft.place=New+York&amp;rft.pub=Cambridge+University+Press&amp;rft.isbn=0-521-00428-4&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFKaiser1994">Kaiser, Gerald (1994), <i>A Friendly Guide to Wavelets</i>, Birkhäuser, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-8176-3711-7" title="Special:BookSources/0-8176-3711-7">0-8176-3711-7</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=A+Friendly+Guide+to+Wavelets&amp;rft.aulast=Kaiser&amp;rft.aufirst=Gerald&amp;rft.au=Kaiser%2C%26%2332%3BGerald&amp;rft.date=1994&amp;rft.pub=Birkh%C3%A4user&amp;rft.isbn=0-8176-3711-7&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation" id="CITEREFKammler2000">Kammler, David (2000), <i>A First Course in Fourier Analysis</i>, Prentice Hall, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-13-578782-3" title="Special:BookSources/0-13-578782-3">0-13-578782-3</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=A+First+Course+in+Fourier+Analysis&amp;rft.aulast=Kammler&amp;rft.aufirst=David&amp;rft.au=Kammler%2C%26%2332%3BDavid&amp;rft.date=2000&amp;rft.pub=Prentice+Hall&amp;rft.isbn=0-13-578782-3&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation" id="CITEREFKatznelson1976">Katznelson, Yitzhak (1976), <i>An introduction to Harmonic Analysis</i>, Dover, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-486-63331-4" title="Special:BookSources/0-486-63331-4">0-486-63331-4</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=An+introduction+to+Harmonic+Analysis&amp;rft.aulast=Katznelson&amp;rft.aufirst=Yitzhak&amp;rft.au=Katznelson%2C%26%2332%3BYitzhak&amp;rft.date=1976&amp;rft.pub=Dover&amp;rft.isbn=0-486-63331-4&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation" id="CITEREFKnapp2001">Knapp, Anthony W. (2001), <a href="http://books.google.com/?id=QCcW1h835pwC" class="external text" rel="nofollow"><i>Representation Theory of Semisimple Groups: An Overview Based on Examples</i></a>, <a href="http://en.wikipedia.org/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-691-09089-4" title="Special:BookSources/978-0-691-09089-4">978-0-691-09089-4</a><span class="printonly">, <a href="http://books.google.com/?id=QCcW1h835pwC" class="external free" rel="nofollow">http://books.google.com/?id=QCcW1h835pwC</a></span></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Representation+Theory+of+Semisimple+Groups%3A+An+Overview+Based+on+Examples&amp;rft.aulast=Knapp&amp;rft.aufirst=Anthony+W.&amp;rft.au=Knapp%2C%26%2332%3BAnthony+W.&amp;rft.date=2001&amp;rft.pub=%5B%5BPrinceton+University+Press%5D%5D&amp;rft.isbn=978-0-691-09089-4&amp;rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3DQCcW1h835pwC&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation" id="CITEREFPinsky2002">Pinsky, Mark (2002), <i>Introduction to Fourier Analysis and Wavelets</i>, Brooks/Cole, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-534-37660-6" title="Special:BookSources/0-534-37660-6">0-534-37660-6</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Introduction+to+Fourier+Analysis+and+Wavelets&amp;rft.aulast=Pinsky&amp;rft.aufirst=Mark&amp;rft.au=Pinsky%2C%26%2332%3BMark&amp;rft.date=2002&amp;rft.pub=Brooks%2FCole&amp;rft.isbn=0-534-37660-6&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation" id="CITEREFPolyaninManzhirov1998">Polyanin, A. D.; Manzhirov, A. V. (1998), <i>Handbook of Integral Equations</i>, Boca Raton: CRC Press, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-8493-2876-4" title="Special:BookSources/0-8493-2876-4">0-8493-2876-4</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Handbook+of+Integral+Equations&amp;rft.aulast=Polyanin&amp;rft.aufirst=A.+D.&amp;rft.au=Polyanin%2C%26%2332%3BA.+D.&amp;rft.au=Manzhirov%2C%26%2332%3BA.+V.&amp;rft.date=1998&amp;rft.place=Boca+Raton&amp;rft.pub=CRC+Press&amp;rft.isbn=0-8493-2876-4&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFRudin1987">Rudin, Walter (1987), <i>Real and Complex Analysis</i> (Third ed.), Singapore: McGraw Hill, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-07-100276-6" title="Special:BookSources/0-07-100276-6">0-07-100276-6</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Real+and+Complex+Analysis&amp;rft.aulast=Rudin&amp;rft.aufirst=Walter&amp;rft.au=Rudin%2C%26%2332%3BWalter&amp;rft.date=1987&amp;rft.edition=Third&amp;rft.place=Singapore&amp;rft.pub=McGraw+Hill&amp;rft.isbn=0-07-100276-6&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFSteinShakarchi2003">Stein, Elias; Shakarchi, Rami (2003), <i>Fourier Analysis: An introduction</i>, Princeton University Press, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-691-11384-X" title="Special:BookSources/0-691-11384-X">0-691-11384-X</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Fourier+Analysis%3A+An+introduction&amp;rft.aulast=Stein&amp;rft.aufirst=Elias&amp;rft.au=Stein%2C%26%2332%3BElias&amp;rft.au=Shakarchi%2C%26%2332%3BRami&amp;rft.date=2003&amp;rft.pub=Princeton+University+Press&amp;rft.isbn=0-691-11384-X&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFSteinWeiss1971"><a href="http://en.wikipedia.org/wiki/Elias_Stein" title="Elias Stein" class="mw-redirect">Stein, Elias</a>; <a href="http://en.wikipedia.org/w/index.php?title=Guido_Weiss&amp;action=edit&amp;redlink=1" class="new" title="Guido Weiss (page does not exist)">Weiss, Guido</a> (1971), <i>Introduction to Fourier Analysis on Euclidean Spaces</i>, Princeton, N.J.: Princeton University Press, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-691-08078-9" title="Special:BookSources/978-0-691-08078-9">978-0-691-08078-9</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Introduction+to+Fourier+Analysis+on+Euclidean+Spaces&amp;rft.aulast=Stein&amp;rft.aufirst=Elias&amp;rft.au=Stein%2C%26%2332%3BElias&amp;rft.au=Weiss%2C%26%2332%3BGuido&amp;rft.date=1971&amp;rft.place=Princeton%2C+N.J.&amp;rft.pub=Princeton+University+Press&amp;rft.isbn=978-0-691-08078-9&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFWilson1995">Wilson, R. G. (1995), <i>Fourier Series and Optical Transform Techniques in Contemporary Optics</i>, New York: Wiley, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0471303577" title="Special:BookSources/0471303577">0471303577</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Fourier+Series+and+Optical+Transform+Techniques+in+Contemporary+Optics&amp;rft.aulast=Wilson&amp;rft.aufirst=R.+G.&amp;rft.au=Wilson%2C%26%2332%3BR.+G.&amp;rft.date=1995&amp;rft.place=New+York&amp;rft.pub=Wiley&amp;rft.isbn=0471303577&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFYosida1968"><a href="http://en.wikipedia.org/wiki/K%C5%8Dsaku_Yosida" title="Kōsaku Yosida">Yosida, K.</a> (1968), <i>Functional Analysis</i>, Springer-Verlag, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/3-540-58654-7" title="Special:BookSources/3-540-58654-7">3-540-58654-7</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Functional+Analysis&amp;rft.aulast=Yosida&amp;rft.aufirst=K.&amp;rft.au=Yosida%2C%26%2332%3BK.&amp;rft.date=1968&amp;rft.pub=Springer-Verlag&amp;rft.isbn=3-540-58654-7&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_transform"><span style="display: none;">&nbsp;</span></span>.</li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_transform&amp;action=edit&amp;section=37" title="Edit section: External links">edit</a>]</span> <span class="mw-headline" id="External_links">External links</span></h2>
<ul>
<li><a href="http://www.thefouriertransform.com/" class="external text" rel="nofollow">Fourier Transform Tutorial</a></li>
<li><a href="http://www.westga.edu/%7Ejhasbun/osp/Fourier.htm" class="external text" rel="nofollow">Fourier Series Applet</a> (Tip: drag magnitude or phase dots up or down to change the wave form).</li>
<li><a href="http://www.dspdimension.com/fftlab/" class="external text" rel="nofollow">Stephan Bernsee's FFTlab</a> (Java Applet)</li>
<li><a href="http://www.academicearth.com/courses/the-fourier-transform-and-its-applications" class="external text" rel="nofollow">Stanford Video Course on the Fourier Transform</a></li>
<li><a href="http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm" class="external text" rel="nofollow">Tables of Integral Transforms</a> at EqWorld: The World of Mathematical Equations.</li>
<li><span class="citation mathworld" id="Reference-Mathworld-Fourier_Transform"><a href="http://en.wikipedia.org/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, "<a href="http://mathworld.wolfram.com/FourierTransform.html" class="external text" rel="nofollow">Fourier Transform</a>" from <a href="http://en.wikipedia.org/wiki/MathWorld" title="MathWorld">MathWorld</a>.</span></li>
<li><a href="http://math.fullerton.edu/mathews/n2003/FourierTransformMod.html" class="external text" rel="nofollow">Fourier Transform Module by John H. Mathews</a></li>
<li><a href="http://www.dspdimension.com/admin/dft-a-pied/" class="external text" rel="nofollow">The DFT “à Pied”: Mastering The Fourier Transform in One Day</a> at The DSP Dimension</li>
<li><a href="http://www.fourier-series.com/f-transform/index.html" class="external text" rel="nofollow">An Interactive Flash Tutorial for the Fourier Transform</a></li>
</ul>


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		<ul>
					<li class="interwiki-am"><a href="http://am.wikipedia.org/wiki/%E1%8B%A8%E1%8D%8E%E1%88%AA%E1%8B%A8%E1%88%AD_%E1%88%BD%E1%8C%8D%E1%8C%8D%E1%88%AD" title="የፎሪየር ሽግግር">አማርኛ</a></li>
					<li class="interwiki-ar"><a href="http://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%88%D9%8A%D9%84_%D9%81%D9%88%D8%B1%D9%8A%D9%8A%D9%87" title="تحويل فورييه">العربية</a></li>
					<li class="interwiki-be-x-old"><a href="http://be-x-old.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B0%D1%9E%D1%82%D0%B2%D0%B0%D1%80%D1%8D%D0%BD%D1%8C%D0%BD%D0%B5_%D0%A4%D1%83%D1%80%27%D0%B5" title="Пераўтварэньне Фур'е">‪Беларуская (тарашкевіца)‬</a></li>
					<li class="interwiki-ca"><a href="http://ca.wikipedia.org/wiki/Transformada_de_Fourier" title="Transformada de Fourier">Català</a></li>
					<li class="interwiki-cs"><a href="http://cs.wikipedia.org/wiki/Fourierova_transformace" title="Fourierova transformace">Česky</a></li>
					<li class="interwiki-da"><a href="http://da.wikipedia.org/wiki/Fouriertransformation" title="Fouriertransformation">Dansk</a></li>
					<li class="interwiki-de"><a href="http://de.wikipedia.org/wiki/Fourier-Transformation" title="Fourier-Transformation">Deutsch</a></li>
					<li class="interwiki-et"><a href="http://et.wikipedia.org/wiki/Fourier%27_teisendus" title="Fourier' teisendus">Eesti</a></li>
					<li class="interwiki-es"><a href="http://es.wikipedia.org/wiki/Transformada_de_Fourier" title="Transformada de Fourier">Español</a></li>
					<li class="interwiki-eo"><a href="http://eo.wikipedia.org/wiki/Konverto_de_Fourier" title="Konverto de Fourier">Esperanto</a></li>
					<li class="interwiki-eu"><a href="http://eu.wikipedia.org/wiki/Fourierren_transformatu" title="Fourierren transformatu">Euskara</a></li>
					<li class="interwiki-fa"><a href="http://fa.wikipedia.org/wiki/%D8%AA%D8%A8%D8%AF%DB%8C%D9%84_%D9%81%D9%88%D8%B1%DB%8C%D9%87" title="تبدیل فوریه">فارسی</a></li>
					<li class="interwiki-fr"><a href="http://fr.wikipedia.org/wiki/Transform%C3%A9e_de_Fourier" title="Transformée de Fourier">Français</a></li>
					<li class="interwiki-gl"><a href="http://gl.wikipedia.org/wiki/Transformada_de_Fourier" title="Transformada de Fourier">Galego</a></li>
					<li class="interwiki-ko"><a href="http://ko.wikipedia.org/wiki/%ED%91%B8%EB%A6%AC%EC%97%90_%EB%B3%80%ED%99%98" title="푸리에 변환">한국어</a></li>
					<li class="interwiki-id"><a href="http://id.wikipedia.org/wiki/Transformasi_Fourier" title="Transformasi Fourier">Bahasa Indonesia</a></li>
					<li class="interwiki-is"><a href="http://is.wikipedia.org/wiki/Fourier%E2%80%93v%C3%B6rpun" title="Fourier–vörpun">Íslenska</a></li>
					<li class="interwiki-it"><a href="http://it.wikipedia.org/wiki/Trasformata_di_Fourier" title="Trasformata di Fourier">Italiano</a></li>
					<li class="interwiki-he"><a href="http://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%9E%D7%A8%D7%AA_%D7%A4%D7%95%D7%A8%D7%99%D7%99%D7%94" title="התמרת פורייה">עברית</a></li>
					<li class="interwiki-lt"><a href="http://lt.wikipedia.org/wiki/Furj%C4%97_transformacija" title="Furjė transformacija">Lietuvių</a></li>
					<li class="interwiki-hu"><a href="http://hu.wikipedia.org/wiki/Fourier-transzform%C3%A1ci%C3%B3" title="Fourier-transzformáció">Magyar</a></li>
					<li class="interwiki-mt"><a href="http://mt.wikipedia.org/wiki/Trasformata_ta%27_Fourier" title="Trasformata ta' Fourier">Malti</a></li>
					<li class="interwiki-mn"><a href="http://mn.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D1%8C%D0%B5_%D1%85%D1%83%D0%B2%D0%B8%D1%80%D0%B3%D0%B0%D0%BB%D1%82" title="Фурье хувиргалт">Монгол</a></li>
					<li class="interwiki-nl"><a href="http://nl.wikipedia.org/wiki/Fouriertransformatie" title="Fouriertransformatie">Nederlands</a></li>
					<li class="interwiki-ja"><a href="http://ja.wikipedia.org/wiki/%E3%83%95%E3%83%BC%E3%83%AA%E3%82%A8%E5%A4%89%E6%8F%9B" title="フーリエ変換">日本語</a></li>
					<li class="interwiki-no"><a href="http://no.wikipedia.org/wiki/Fouriertransformasjon" title="Fouriertransformasjon">‪Norsk (bokmål)‬</a></li>
					<li class="interwiki-nn"><a href="http://nn.wikipedia.org/wiki/Fouriertransformasjon" title="Fouriertransformasjon">‪Norsk (nynorsk)‬</a></li>
					<li class="interwiki-pl"><a href="http://pl.wikipedia.org/wiki/Transformacja_Fouriera" title="Transformacja Fouriera">Polski</a></li>
					<li class="interwiki-pt"><a href="http://pt.wikipedia.org/wiki/Transformada_de_Fourier" title="Transformada de Fourier">Português</a></li>
					<li class="interwiki-ro"><a href="http://ro.wikipedia.org/wiki/Transformata_Fourier" title="Transformata Fourier">Română</a></li>
					<li class="interwiki-ru"><a href="http://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5_%D0%A4%D1%83%D1%80%D1%8C%D0%B5" title="Преобразование Фурье">Русский</a></li>
					<li class="interwiki-sq"><a href="http://sq.wikipedia.org/wiki/Transformimi_i_Furierit" title="Transformimi i Furierit">Shqip</a></li>
					<li class="interwiki-simple"><a href="http://simple.wikipedia.org/wiki/Fourier_transform" title="Fourier transform">Simple English</a></li>
					<li class="interwiki-sk"><a href="http://sk.wikipedia.org/wiki/Fourierova_transform%C3%A1cia" title="Fourierova transformácia">Slovenčina</a></li>
					<li class="interwiki-sr"><a href="http://sr.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D0%B8%D1%98%D0%B5%D0%BE%D0%B2_%D1%80%D0%B5%D0%B4" title="Фуријеов ред">Српски / Srpski</a></li>
					<li class="interwiki-su"><a href="http://su.wikipedia.org/wiki/Transformasi_Fourier" title="Transformasi Fourier">Basa Sunda</a></li>
					<li class="interwiki-fi"><a href="http://fi.wikipedia.org/wiki/Fourier%27n_muunnos" title="Fourier'n muunnos">Suomi</a></li>
					<li class="interwiki-sv"><a href="http://sv.wikipedia.org/wiki/Fouriertransform" title="Fouriertransform">Svenska</a></li>
					<li class="interwiki-ta"><a href="http://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%82%E0%AE%B0%E0%AE%BF%E0%AE%AF%E0%AF%87_%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81" title="வூரியே மாற்று">தமிழ்</a></li>
					<li class="interwiki-th"><a href="http://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%9B%E0%B8%A5%E0%B8%87%E0%B8%9F%E0%B8%B9%E0%B8%A3%E0%B8%B4%E0%B9%80%E0%B8%A2%E0%B8%A3%E0%B9%8C" title="การแปลงฟูริเยร์">ไทย</a></li>
					<li class="interwiki-tr"><a href="http://tr.wikipedia.org/wiki/Fourier_d%C3%B6n%C3%BC%C5%9F%C3%BCm%C3%BC" title="Fourier dönüşümü">Türkçe</a></li>
					<li class="interwiki-uk"><a href="http://uk.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B5%D1%82%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F_%D0%A4%D1%83%D1%80%27%D1%94" title="Перетворення Фур'є">Українська</a></li>
					<li class="interwiki-vi"><a href="http://vi.wikipedia.org/wiki/Bi%E1%BA%BFn_%C4%91%E1%BB%95i_Fourier" title="Biến đổi Fourier">Tiếng Việt</a></li>
					<li class="interwiki-zh"><a href="http://zh.wikipedia.org/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="傅里叶变换">中文</a></li>
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